Theorem dfon2lem7 36759

Description: Lemma for dfon2 36762. All elements of a new ordinal are new ordinals. (Contributed by Scott Fenton, 25-Feb-2011.)

Hypothesis

Ref	Expression
dfon2lem7.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
dfon2lem7	⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (𝐵 ∈ 𝐴 → ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem dfon2lem7

Dummy variables 𝑧 𝑤 𝑠 𝑡 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elequ1 2154	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑧 → (𝑡 ∈ 𝑡 ↔ 𝑧 ∈ 𝑡))
2		elequ2 2162	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑧 → (𝑧 ∈ 𝑡 ↔ 𝑧 ∈ 𝑧))
3	1, 2	bitrd 282	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑧 → (𝑡 ∈ 𝑡 ↔ 𝑧 ∈ 𝑧))
4	3	notbid 321	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑧 → (¬ 𝑡 ∈ 𝑡 ↔ ¬ 𝑧 ∈ 𝑧))
5	4	cbvralvw 3244	. . . . . . . . . . . . 13 ⊢ (∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡 ↔ ∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
6	5	biimpi 219	. . . . . . . . . . . 12 ⊢ (∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡 → ∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
7	6	ralimi 3103	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡 → ∀𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
8		untuni 36682	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ¬ 𝑧 ∈ 𝑧 ↔ ∀𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
9	7, 8	sylibr 237	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡 → ∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ¬ 𝑧 ∈ 𝑧)
10		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
11		sseq1 3938	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
12		treq 5223	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → (Tr 𝑤 ↔ Tr 𝑥))
13		raleq 3320	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → (∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) ↔ ∀𝑡 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡)))
14	11, 12, 13	3anbi123d 1465	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → ((𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡)) ↔ (𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑡 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))))
15	10, 14	elab 3636	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ↔ (𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑡 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡)))
16		vex 3458	. . . . . . . . . . . . . . . 16 ⊢ 𝑡 ∈ V
17		dfon2lem3 36755	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ V → (∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) → (Tr 𝑡 ∧ ∀𝑢 ∈ 𝑡 ¬ 𝑢 ∈ 𝑢)))
18	16, 17	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) → (Tr 𝑡 ∧ ∀𝑢 ∈ 𝑡 ¬ 𝑢 ∈ 𝑢))
19	18	simprd 501	. . . . . . . . . . . . . 14 ⊢ (∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) → ∀𝑢 ∈ 𝑡 ¬ 𝑢 ∈ 𝑢)
20		untelirr 36681	. . . . . . . . . . . . . 14 ⊢ (∀𝑢 ∈ 𝑡 ¬ 𝑢 ∈ 𝑢 → ¬ 𝑡 ∈ 𝑡)
21	19, 20	syl 18	. . . . . . . . . . . . 13 ⊢ (∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) → ¬ 𝑡 ∈ 𝑡)
22	21	ralimi 3103	. . . . . . . . . . . 12 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) → ∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡)
23	22	3ad2ant3 1153	. . . . . . . . . . 11 ⊢ ((𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑡 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡)) → ∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡)
24	15, 23	sylbi 220	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → ∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡)
25	9, 24	mprg 3086	. . . . . . . . 9 ⊢ ∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ¬ 𝑧 ∈ 𝑧
26		untelirr 36681	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ¬ 𝑧 ∈ 𝑧 → ¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))})
27		psseq2 4021	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 = 𝑢 → (𝑦 ⊊ 𝑡 ↔ 𝑦 ⊊ 𝑢))
28	27	anbi1d 643	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑢 → ((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) ↔ (𝑦 ⊊ 𝑢 ∧ Tr 𝑦)))
29		elequ2 2162	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 = 𝑢 → (𝑦 ∈ 𝑡 ↔ 𝑦 ∈ 𝑢))
30	28, 29	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 = 𝑢 → (((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) ↔ ((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢)))
31	30	albidv 1955	. . . . . . . . . . . . . . 15 ⊢ (𝑡 = 𝑢 → (∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) ↔ ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢)))
32	31	cbvralvw 3244	. . . . . . . . . . . . . 14 ⊢ (∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) ↔ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))
33	32	3anbi3i 1177	. . . . . . . . . . . . 13 ⊢ ((𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡)) ↔ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢)))
34	33	abbii 2830	. . . . . . . . . . . 12 ⊢ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} = {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}
35	34	unieqi 4882	. . . . . . . . . . 11 ⊢ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}
36	35	eleq2i 2856	. . . . . . . . . 10 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ↔ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))})
37	26, 36	sylnib 331	. . . . . . . . 9 ⊢ (∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ¬ 𝑧 ∈ 𝑧 → ¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))})
38	25, 37	ax-mp 5	. . . . . . . 8 ⊢ ¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}
39		dfon2lem7.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ V
40		dfon2lem2 36754	. . . . . . . . . 10 ⊢ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴
41	39, 40	ssexi 5289	. . . . . . . . 9 ⊢ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ V
42	41	snss 4745	. . . . . . . 8 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))} ↔ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))})
43	38, 42	mtbi 325	. . . . . . 7 ⊢ ¬ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}
44	43	intnan 492	. . . . . 6 ⊢ ¬ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))} ∧ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))})
45		df-suc 6392	. . . . . . . 8 ⊢ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} = (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∪ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}})
46	45	sseq1i 3941	. . . . . . 7 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))} ↔ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∪ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}}) ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))})
47		unss 4191	. . . . . . 7 ⊢ ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))} ∧ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}) ↔ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∪ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}}) ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))})
48	46, 47	bitr4i 281	. . . . . 6 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))} ↔ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))} ∧ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}))
49	44, 48	mtbir 326	. . . . 5 ⊢ ¬ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}
50	41	snss 4745	. . . . . 6 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ 𝐴 ↔ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}} ⊆ 𝐴)
51	45	sseq1i 3941	. . . . . . . . 9 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 ↔ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∪ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}}) ⊆ 𝐴)
52		unss 4191	. . . . . . . . 9 ⊢ ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 ∧ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}} ⊆ 𝐴) ↔ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∪ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}}) ⊆ 𝐴)
53	51, 52	bitr4i 281	. . . . . . . 8 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 ↔ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 ∧ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}} ⊆ 𝐴))
54		dfon2lem1 36753	. . . . . . . . . . . 12 ⊢ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}
55		suctr 6475	. . . . . . . . . . . 12 ⊢ (Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))})
56	54, 55	ax-mp 5	. . . . . . . . . . 11 ⊢ Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}
57		vex 3458	. . . . . . . . . . . . . 14 ⊢ 𝑢 ∈ V
58	57	elsuc 6459	. . . . . . . . . . . . 13 ⊢ (𝑢 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ↔ (𝑢 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∨ 𝑢 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}))
59		eluni2 4874	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ↔ ∃𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}𝑢 ∈ 𝑥)
60		nfa1 2191	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢)
61	31	rspccv 3577	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) → (𝑢 ∈ 𝑥 → ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢)))
62		psseq1 4020	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑥 → (𝑦 ⊊ 𝑢 ↔ 𝑥 ⊊ 𝑢))
63		treq 5223	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = 𝑥 → (Tr 𝑦 ↔ Tr 𝑥))
64	62, 63	anbi12d 644	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑥 → ((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) ↔ (𝑥 ⊊ 𝑢 ∧ Tr 𝑥)))
65		elequ1 2154	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝑢 ↔ 𝑥 ∈ 𝑢))
66	64, 65	imbi12d 347	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢) ↔ ((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢)))
67	66	cbvalvw 2071	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢) ↔ ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))
68	61, 67	imbitrdi 254	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) → (𝑢 ∈ 𝑥 → ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢)))
69	68	3ad2ant3 1153	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑡 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡)) → (𝑢 ∈ 𝑥 → ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢)))
70	15, 69	sylbi 220	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (𝑢 ∈ 𝑥 → ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢)))
71	60, 70	rexlimi 3266	. . . . . . . . . . . . . . 15 ⊢ (∃𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}𝑢 ∈ 𝑥 → ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))
72	59, 71	sylbi 220	. . . . . . . . . . . . . 14 ⊢ (𝑢 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))
73		psseq1 4020	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑧 → (𝑦 ⊊ 𝑢 ↔ 𝑧 ⊊ 𝑢))
74		treq 5223	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 = 𝑧 → (Tr 𝑦 ↔ Tr 𝑧))
75	73, 74	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 → ((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) ↔ (𝑧 ⊊ 𝑢 ∧ Tr 𝑧)))
76		elequ1 2154	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝑢 ↔ 𝑧 ∈ 𝑢))
77	75, 76	imbi12d 347	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → (((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢) ↔ ((𝑧 ⊊ 𝑢 ∧ Tr 𝑧) → 𝑧 ∈ 𝑢)))
78	77	cbvalvw 2071	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢) ↔ ∀𝑧((𝑧 ⊊ 𝑢 ∧ Tr 𝑧) → 𝑧 ∈ 𝑢))
79	61, 78	imbitrdi 254	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) → (𝑢 ∈ 𝑥 → ∀𝑧((𝑧 ⊊ 𝑢 ∧ Tr 𝑧) → 𝑧 ∈ 𝑢)))
80	79	3ad2ant3 1153	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑡 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡)) → (𝑢 ∈ 𝑥 → ∀𝑧((𝑧 ⊊ 𝑢 ∧ Tr 𝑧) → 𝑧 ∈ 𝑢)))
81	15, 80	sylbi 220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (𝑢 ∈ 𝑥 → ∀𝑧((𝑧 ⊊ 𝑢 ∧ Tr 𝑧) → 𝑧 ∈ 𝑢)))
82	81	rexlimiv 3160	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}𝑢 ∈ 𝑥 → ∀𝑧((𝑧 ⊊ 𝑢 ∧ Tr 𝑧) → 𝑧 ∈ 𝑢))
83	59, 82	sylbi 220	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → ∀𝑧((𝑧 ⊊ 𝑢 ∧ Tr 𝑧) → 𝑧 ∈ 𝑢))
84	83	rgen 3082	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑢 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑧((𝑧 ⊊ 𝑢 ∧ Tr 𝑧) → 𝑧 ∈ 𝑢)
85		dfon2lem6 36758	. . . . . . . . . . . . . . . 16 ⊢ ((Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∧ ∀𝑢 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑧((𝑧 ⊊ 𝑢 ∧ Tr 𝑧) → 𝑧 ∈ 𝑢)) → ∀𝑥((𝑥 ⊊ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∧ Tr 𝑥) → 𝑥 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}))
86	54, 84, 85	mp2an 705	. . . . . . . . . . . . . . 15 ⊢ ∀𝑥((𝑥 ⊊ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∧ Tr 𝑥) → 𝑥 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))})
87		psseq2 4021	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (𝑥 ⊊ 𝑢 ↔ 𝑥 ⊊ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}))
88	87	anbi1d 643	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → ((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) ↔ (𝑥 ⊊ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∧ Tr 𝑥)))
89		eleq2 2853	. . . . . . . . . . . . . . . . 17 ⊢ (𝑢 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}))
90	88, 89	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑢 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢) ↔ ((𝑥 ⊊ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∧ Tr 𝑥) → 𝑥 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))})))
91	90	albidv 1955	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢) ↔ ∀𝑥((𝑥 ⊊ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∧ Tr 𝑥) → 𝑥 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))})))
92	86, 91	mpbiri 261	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))
93	72, 92	jaoi 871	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∨ 𝑢 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}) → ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))
94	58, 93	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))
95	94	rgen 3082	. . . . . . . . . . 11 ⊢ ∀𝑢 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢)
96	41	sucex 7852	. . . . . . . . . . . . 13 ⊢ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ V
97		sseq1 3938	. . . . . . . . . . . . . 14 ⊢ (𝑠 = suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (𝑠 ⊆ 𝐴 ↔ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴))
98		treq 5223	. . . . . . . . . . . . . 14 ⊢ (𝑠 = suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (Tr 𝑠 ↔ Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}))
99		raleq 3320	. . . . . . . . . . . . . 14 ⊢ (𝑠 = suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (∀𝑢 ∈ 𝑠 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢) ↔ ∀𝑢 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢)))
100	97, 98, 99	3anbi123d 1465	. . . . . . . . . . . . 13 ⊢ (𝑠 = suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → ((𝑠 ⊆ 𝐴 ∧ Tr 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢)) ↔ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 ∧ Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∧ ∀𝑢 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))))
101	96, 100	elab 3636	. . . . . . . . . . . 12 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ Tr 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))} ↔ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 ∧ Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∧ ∀𝑢 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢)))
102		elssuni 4902	. . . . . . . . . . . 12 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ Tr 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))} → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ Tr 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))})
103	101, 102	sylbir 238	. . . . . . . . . . 11 ⊢ ((suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 ∧ Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∧ ∀𝑢 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢)) → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ Tr 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))})
104	56, 95, 103	mp3an23 1483	. . . . . . . . . 10 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ Tr 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))})
105		sseq1 3938	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑤 → (𝑠 ⊆ 𝐴 ↔ 𝑤 ⊆ 𝐴))
106		treq 5223	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑤 → (Tr 𝑠 ↔ Tr 𝑤))
107		raleq 3320	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑤 → (∀𝑢 ∈ 𝑠 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢) ↔ ∀𝑢 ∈ 𝑤 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢)))
108		psseq1 4020	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ⊊ 𝑢 ↔ 𝑦 ⊊ 𝑢))
109		treq 5223	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
110	108, 109	anbi12d 644	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → ((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) ↔ (𝑦 ⊊ 𝑢 ∧ Tr 𝑦)))
111		elequ1 2154	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑢 ↔ 𝑦 ∈ 𝑢))
112	110, 111	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢) ↔ ((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢)))
113	112	cbvalvw 2071	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢) ↔ ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))
114	113	ralbii 3112	. . . . . . . . . . . . . 14 ⊢ (∀𝑢 ∈ 𝑤 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢) ↔ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))
115	107, 114	bitrdi 290	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑤 → (∀𝑢 ∈ 𝑠 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢) ↔ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢)))
116	105, 106, 115	3anbi123d 1465	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑤 → ((𝑠 ⊆ 𝐴 ∧ Tr 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢)) ↔ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))))
117	116	cbvabv 2834	. . . . . . . . . . 11 ⊢ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ Tr 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))} = {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}
118	117	unieqi 4882	. . . . . . . . . 10 ⊢ ∪ {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ Tr 𝑠 ∧ ∀𝑢 ∈ 𝑠 ∀𝑥((𝑥 ⊊ 𝑢 ∧ Tr 𝑥) → 𝑥 ∈ 𝑢))} = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}
119	104, 118	sseqtrdi 3953	. . . . . . . . 9 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))})
120	119	a1i 11	. . . . . . . 8 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}))
121	53, 120	biimtrrid 246	. . . . . . 7 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 ∧ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}} ⊆ 𝐴) → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}))
122	40, 121	mpani 709	. . . . . 6 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → ({∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}} ⊆ 𝐴 → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}))
123	50, 122	biimtrid 245	. . . . 5 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ 𝐴 → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑢 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑢 ∧ Tr 𝑦) → 𝑦 ∈ 𝑢))}))
124	49, 123	mtoi 202	. . . 4 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → ¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ 𝐴)
125		psseq1 4020	. . . . . . . 8 ⊢ (𝑥 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (𝑥 ⊊ 𝐴 ↔ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊊ 𝐴))
126		treq 5223	. . . . . . . 8 ⊢ (𝑥 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (Tr 𝑥 ↔ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}))
127	125, 126	anbi12d 644	. . . . . . 7 ⊢ (𝑥 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → ((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) ↔ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊊ 𝐴 ∧ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))})))
128		eleq1 2852	. . . . . . 7 ⊢ (𝑥 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (𝑥 ∈ 𝐴 ↔ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ 𝐴))
129	127, 128	imbi12d 347	. . . . . 6 ⊢ (𝑥 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ↔ ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊊ 𝐴 ∧ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}) → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ 𝐴)))
130	41, 129	spcv 3563	. . . . 5 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊊ 𝐴 ∧ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}) → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ 𝐴))
131	54, 130	mpan2i 710	. . . 4 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊊ 𝐴 → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ∈ 𝐴))
132	124, 131	mtod 201	. . 3 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → ¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊊ 𝐴)
133		dfpss2 4018	. . . . 5 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊊ 𝐴 ↔ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 ∧ ¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} = 𝐴))
134	133	biimpri 231	. . . 4 ⊢ ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊆ 𝐴 ∧ ¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} = 𝐴) → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊊ 𝐴)
135	40, 134	mpan 703	. . 3 ⊢ (¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} = 𝐴 → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ⊊ 𝐴)
136	132, 135	nsyl2 142	. 2 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} = 𝐴)
137		eluni2 4874	. . . . 5 ⊢ (𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ↔ ∃𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}𝑧 ∈ 𝑥)
138		psseq2 4021	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑧 → (𝑦 ⊊ 𝑡 ↔ 𝑦 ⊊ 𝑧))
139	138	anbi1d 643	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → ((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) ↔ (𝑦 ⊊ 𝑧 ∧ Tr 𝑦)))
140		elequ2 2162	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → (𝑦 ∈ 𝑡 ↔ 𝑦 ∈ 𝑧))
141	139, 140	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → (((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) ↔ ((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧)))
142	141	albidv 1955	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑧 → (∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) ↔ ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧)))
143	142	cbvralvw 3244	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) ↔ ∀𝑧 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧))
144	13, 143	bitrdi 290	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → (∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡) ↔ ∀𝑧 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧)))
145	11, 12, 144	3anbi123d 1465	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → ((𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡)) ↔ (𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧))))
146	10, 145	elab 3636	. . . . . . 7 ⊢ (𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} ↔ (𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧)))
147		rsp 3254	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧) → (𝑧 ∈ 𝑥 → ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧)))
148	147	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑧 ∈ 𝑥 ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧)) → (𝑧 ∈ 𝑥 → ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧)))
149	146, 148	sylbi 220	. . . . . 6 ⊢ (𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → (𝑧 ∈ 𝑥 → ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧)))
150	149	rexlimiv 3160	. . . . 5 ⊢ (∃𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}𝑧 ∈ 𝑥 → ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧))
151	137, 150	sylbi 220	. . . 4 ⊢ (𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} → ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧))
152	151	rgen 3082	. . 3 ⊢ ∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧)
153		raleq 3320	. . 3 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} = 𝐴 → (∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))}∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧) ↔ ∀𝑧 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧)))
154	152, 153	mpbii 236	. 2 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ∀𝑦((𝑦 ⊊ 𝑡 ∧ Tr 𝑦) → 𝑦 ∈ 𝑡))} = 𝐴 → ∀𝑧 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧))
155		psseq2 4021	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑦 ⊊ 𝑧 ↔ 𝑦 ⊊ 𝐵))
156	155	anbi1d 643	. . . . 5 ⊢ (𝑧 = 𝐵 → ((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) ↔ (𝑦 ⊊ 𝐵 ∧ Tr 𝑦)))
157		eleq2 2853	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝑦 ∈ 𝑧 ↔ 𝑦 ∈ 𝐵))
158	156, 157	imbi12d 347	. . . 4 ⊢ (𝑧 = 𝐵 → (((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧) ↔ ((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)))
159	158	albidv 1955	. . 3 ⊢ (𝑧 = 𝐵 → (∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧) ↔ ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)))
160	159	rspccv 3577	. 2 ⊢ (∀𝑧 ∈ 𝐴 ∀𝑦((𝑦 ⊊ 𝑧 ∧ Tr 𝑦) → 𝑦 ∈ 𝑧) → (𝐵 ∈ 𝐴 → ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)))
161	136, 154, 160	3syl 19	1 ⊢ (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (𝐵 ∈ 𝐴 → ∀𝑦((𝑦 ⊊ 𝐵 ∧ Tr 𝑦) → 𝑦 ∈ 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ⊊ wpss 4016   ∪ cun 4152  {csn 4588  ∪ cuni 4870  Tr wtr 5216  suc csuc 6388

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396  ax-un 7781

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-pw 4562  df-sn 4589  df-pr 4591  df-uni 4871  df-iun 4956  df-tr 5217  df-suc 6392

This theorem is used by:  dfon2lem8  36760  dfon2  36762