Theorem dfon2lem3 33480

Description: Lemma for dfon2 33487. All sets satisfying the new definition are transitive and untangled. (Contributed by Scott Fenton, 25-Feb-2011.)

Assertion

Ref	Expression
dfon2lem3	⊢ (𝐴 ∈ 𝑉 → (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧)))

Distinct variable group:   𝑥,𝐴,𝑧

Allowed substitution hints:   𝑉(𝑥,𝑧)

Proof of Theorem dfon2lem3

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

Step	Hyp	Ref	Expression
1		untelirr 33362	. . . . 5 ⊢ (∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑧 ∈ 𝑧 → ¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)})
2		eluni2 4823	. . . . . 6 ⊢ (𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ↔ ∃𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}𝑧 ∈ 𝑥)
3		vex 3412	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
4		sseq1 3926	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
5		treq 5167	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (Tr 𝑤 ↔ Tr 𝑥))
6		raleq 3319	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡 ↔ ∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡))
7	4, 5, 6	3anbi123d 1438	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → ((𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡) ↔ (𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡)))
8	3, 7	elab 3587	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ↔ (𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡))
9		elequ1 2117	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑧 → (𝑡 ∈ 𝑡 ↔ 𝑧 ∈ 𝑡))
10		elequ2 2125	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑧 → (𝑧 ∈ 𝑡 ↔ 𝑧 ∈ 𝑧))
11	9, 10	bitrd 282	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → (𝑡 ∈ 𝑡 ↔ 𝑧 ∈ 𝑧))
12	11	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → (¬ 𝑡 ∈ 𝑡 ↔ ¬ 𝑧 ∈ 𝑧))
13	12	cbvralvw 3358	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡 ↔ ∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
14	13	biimpi 219	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡 → ∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
15	14	3ad2ant3 1137	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡) → ∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
16	8, 15	sylbi 220	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → ∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
17		rsp 3127	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧 → (𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑧))
18	16, 17	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → (𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑧))
19	18	rexlimiv 3199	. . . . . 6 ⊢ (∃𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑧)
20	2, 19	sylbi 220	. . . . 5 ⊢ (𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → ¬ 𝑧 ∈ 𝑧)
21	1, 20	mprg 3075	. . . 4 ⊢ ¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}
22		dfon2lem2 33479	. . . . 5 ⊢ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴
23		dfpss2 4000	. . . . . 6 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊊ 𝐴 ↔ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 ∧ ¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} = 𝐴))
24		dfon2lem1 33478	. . . . . . 7 ⊢ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}
25		ssexg 5216	. . . . . . . . . 10 ⊢ ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ V)
26	22, 25	mpan 690	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ V)
27		psseq1 4002	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → (𝑥 ⊊ 𝐴 ↔ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊊ 𝐴))
28		treq 5167	. . . . . . . . . . . . 13 ⊢ (𝑥 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → (Tr 𝑥 ↔ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
29	27, 28	anbi12d 634	. . . . . . . . . . . 12 ⊢ (𝑥 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → ((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) ↔ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊊ 𝐴 ∧ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)})))
30		eleq1 2825	. . . . . . . . . . . 12 ⊢ (𝑥 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → (𝑥 ∈ 𝐴 ↔ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴))
31	29, 30	imbi12d 348	. . . . . . . . . . 11 ⊢ (𝑥 = ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → (((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) ↔ ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊊ 𝐴 ∧ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}) → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴)))
32	31	spcgv 3511	. . . . . . . . . 10 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ V → (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊊ 𝐴 ∧ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}) → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴)))
33	32	imp 410	. . . . . . . . 9 ⊢ ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ V ∧ ∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴)) → ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊊ 𝐴 ∧ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}) → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴))
34	26, 33	sylan 583	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴)) → ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊊ 𝐴 ∧ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}) → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴))
35		snssi 4721	. . . . . . . . . 10 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴 → {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}} ⊆ 𝐴)
36		unss 4098	. . . . . . . . . . 11 ⊢ ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 ∧ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}} ⊆ 𝐴) ↔ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∪ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}}) ⊆ 𝐴)
37		df-suc 6219	. . . . . . . . . . . 12 ⊢ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} = (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∪ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}})
38	37	sseq1i 3929	. . . . . . . . . . 11 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 ↔ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∪ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}}) ⊆ 𝐴)
39	36, 38	sylbb2 241	. . . . . . . . . 10 ⊢ ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 ∧ {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}} ⊆ 𝐴) → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴)
40	22, 35, 39	sylancr 590	. . . . . . . . 9 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴 → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴)
41		suctr 6296	. . . . . . . . . . . . 13 ⊢ (Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)})
42	24, 41	ax-mp 5	. . . . . . . . . . . 12 ⊢ Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}
43		untuni 33363	. . . . . . . . . . . . . 14 ⊢ (∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑧 ∈ 𝑧 ↔ ∀𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
44	43, 16	mprgbir 3076	. . . . . . . . . . . . 13 ⊢ ∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑧 ∈ 𝑧
45		nfv 1922	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡 𝑤 ⊆ 𝐴
46		nfv 1922	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡Tr 𝑤
47		nfra1 3140	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡
48	45, 46, 47	nf3an 1909	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡(𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)
49	48	nfab 2910	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡{𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}
50	49	nfuni 4826	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}
51	50	untsucf 33364	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑧 ∈ 𝑧 → ∀𝑡 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑡 ∈ 𝑡)
52	44, 51	ax-mp 5	. . . . . . . . . . . 12 ⊢ ∀𝑡 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑡 ∈ 𝑡
53		sseq1 3926	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → (𝑧 ⊆ 𝐴 ↔ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴))
54		treq 5167	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → (Tr 𝑧 ↔ Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
55		nfcv 2904	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡𝑧
56	50	nfsuc 6284	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡 suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}
57	55, 56	raleqf 3309	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → (∀𝑡 ∈ 𝑧 ¬ 𝑡 ∈ 𝑡 ↔ ∀𝑡 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑡 ∈ 𝑡))
58	53, 54, 57	3anbi123d 1438	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → ((𝑧 ⊆ 𝐴 ∧ Tr 𝑧 ∧ ∀𝑡 ∈ 𝑧 ¬ 𝑡 ∈ 𝑡) ↔ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 ∧ Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∧ ∀𝑡 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑡 ∈ 𝑡)))
59		sseq1 3926	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑤 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴))
60		treq 5167	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (Tr 𝑤 ↔ Tr 𝑧))
61		raleq 3319	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡 ↔ ∀𝑡 ∈ 𝑧 ¬ 𝑡 ∈ 𝑡))
62	59, 60, 61	3anbi123d 1438	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑧 → ((𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡) ↔ (𝑧 ⊆ 𝐴 ∧ Tr 𝑧 ∧ ∀𝑡 ∈ 𝑧 ¬ 𝑡 ∈ 𝑡)))
63	62	cbvabv 2811	. . . . . . . . . . . . . . 15 ⊢ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} = {𝑧 ∣ (𝑧 ⊆ 𝐴 ∧ Tr 𝑧 ∧ ∀𝑡 ∈ 𝑧 ¬ 𝑡 ∈ 𝑡)}
64	58, 63	elab2g 3589	. . . . . . . . . . . . . 14 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ V → (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ↔ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 ∧ Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∧ ∀𝑡 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑡 ∈ 𝑡)))
65	64	biimprd 251	. . . . . . . . . . . . 13 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ V → ((suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 ∧ Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∧ ∀𝑡 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑡 ∈ 𝑡) → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
66		sucexg 7589	. . . . . . . . . . . . 13 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴 → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ V)
67	65, 66	syl11 33	. . . . . . . . . . . 12 ⊢ ((suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 ∧ Tr suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∧ ∀𝑡 ∈ suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑡 ∈ 𝑡) → (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴 → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
68	42, 52, 67	mp3an23 1455	. . . . . . . . . . 11 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 → (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴 → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
69	68	com12 32	. . . . . . . . . 10 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴 → (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
70		elssuni 4851	. . . . . . . . . . 11 ⊢ (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)})
71		sucssel 6305	. . . . . . . . . . 11 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴 → (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
72	70, 71	syl5 34	. . . . . . . . . 10 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴 → (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
73	69, 72	syld 47	. . . . . . . . 9 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴 → (suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
74	40, 73	mpd 15	. . . . . . . 8 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴 → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)})
75	34, 74	syl6 35	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴)) → ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊊ 𝐴 ∧ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}) → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
76	24, 75	mpan2i 697	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴)) → (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊊ 𝐴 → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
77	23, 76	syl5bir 246	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴)) → ((∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴 ∧ ¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} = 𝐴) → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
78	22, 77	mpani 696	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴)) → (¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} = 𝐴 → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}))
79	21, 78	mt3i 151	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴)) → ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} = 𝐴)
80	24, 44	pm3.2i 474	. . . 4 ⊢ (Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∧ ∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑧 ∈ 𝑧)
81		treq 5167	. . . . 5 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} = 𝐴 → (Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ↔ Tr 𝐴))
82		raleq 3319	. . . . 5 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} = 𝐴 → (∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑧 ∈ 𝑧 ↔ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧))
83	81, 82	anbi12d 634	. . . 4 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} = 𝐴 → ((Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∧ ∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑧 ∈ 𝑧) ↔ (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧)))
84	80, 83	mpbii 236	. . 3 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} = 𝐴 → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧))
85	79, 84	syl 17	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴)) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧))
86	85	ex 416	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1089  ∀wal 1541   = wceq 1543   ∈ wcel 2110  {cab 2714  ∀wral 3061  ∃wrex 3062  Vcvv 3408   ∪ cun 3864   ⊆ wss 3866   ⊊ wpss 3867  {csn 4541  ∪ cuni 4819  Tr wtr 5161  suc csuc 6215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-pw 4515  df-sn 4542  df-pr 4544  df-uni 4820  df-iun 4906  df-tr 5162  df-suc 6219

This theorem is referenced by:  dfon2lem4  33481  dfon2lem5  33482  dfon2lem7  33484  dfon2lem8  33485  dfon2lem9  33486  dfon2  33487