Theorem dfon2lem3 35787

Description: Lemma for dfon2 35794. 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 35709	. . . . 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 4910	. . . . . 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 3483	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
4		sseq1 4008	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑤 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐴))
5		treq 5266	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (Tr 𝑤 ↔ Tr 𝑥))
6		raleq 3322	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡 ↔ ∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡))
7	4, 5, 6	3anbi123d 1437	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → ((𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡) ↔ (𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡)))
8	3, 7	elab 3678	. . . . . . . . 9 ⊢ (𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ↔ (𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡))
9		elequ1 2114	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑧 → (𝑡 ∈ 𝑡 ↔ 𝑧 ∈ 𝑡))
10		elequ2 2122	. . . . . . . . . . . . . 14 ⊢ (𝑡 = 𝑧 → (𝑧 ∈ 𝑡 ↔ 𝑧 ∈ 𝑧))
11	9, 10	bitrd 279	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑧 → (𝑡 ∈ 𝑡 ↔ 𝑧 ∈ 𝑧))
12	11	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑧 → (¬ 𝑡 ∈ 𝑡 ↔ ¬ 𝑧 ∈ 𝑧))
13	12	cbvralvw 3236	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡 ↔ ∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
14	13	biimpi 216	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡 → ∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
15	14	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ Tr 𝑥 ∧ ∀𝑡 ∈ 𝑥 ¬ 𝑡 ∈ 𝑡) → ∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
16	8, 15	sylbi 217	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → ∀𝑧 ∈ 𝑥 ¬ 𝑧 ∈ 𝑧)
17		rsp 3246	. . . . . . . 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 3147	. . . . . 6 ⊢ (∃𝑥 ∈ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑧)
20	2, 19	sylbi 217	. . . . 5 ⊢ (𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} → ¬ 𝑧 ∈ 𝑧)
21	1, 20	mprg 3066	. . . 4 ⊢ ¬ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}
22		dfon2lem2 35786	. . . . 5 ⊢ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴
23		dfpss2 4087	. . . . . 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 35785	. . . . . . 7 ⊢ Tr ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}
25		ssexg 5322	. . . . . . . . . 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 4089	. . . . . . . . . . . . 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 5266	. . . . . . . . . . . . 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 632	. . . . . . . . . . . 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 2828	. . . . . . . . . . . 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 344	. . . . . . . . . . 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 3595	. . . . . . . . . 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 406	. . . . . . . . 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 580	. . . . . . . 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 4807	. . . . . . . . . 10 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴 → {∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}} ⊆ 𝐴)
36		unss 4189	. . . . . . . . . . 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 6389	. . . . . . . . . . . 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 4011	. . . . . . . . . . 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 238	. . . . . . . . . 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 587	. . . . . . . . 9 ⊢ (∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ∈ 𝐴 → suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ⊆ 𝐴)
41		suctr 6469	. . . . . . . . . . . . 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 35710	. . . . . . . . . . . . . 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 3067	. . . . . . . . . . . . 13 ⊢ ∀𝑧 ∈ ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)} ¬ 𝑧 ∈ 𝑧
45		nfv 1913	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡 𝑤 ⊆ 𝐴
46		nfv 1913	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡Tr 𝑤
47		nfra1 3283	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡
48	45, 46, 47	nf3an 1900	. . . . . . . . . . . . . . . 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 4913	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}
51	50	untsucf 35711	. . . . . . . . . . . . 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 4008	. . . . . . . . . . . . . . . 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 5266	. . . . . . . . . . . . . . . 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 6455	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡 suc ∪ {𝑤 ∣ (𝑤 ⊆ 𝐴 ∧ Tr 𝑤 ∧ ∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡)}
57	55, 56	raleqf 3352	. . . . . . . . . . . . . . . 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 1437	. . . . . . . . . . . . . . 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 4008	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (𝑤 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴))
60		treq 5266	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (Tr 𝑤 ↔ Tr 𝑧))
61		raleq 3322	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑧 → (∀𝑡 ∈ 𝑤 ¬ 𝑡 ∈ 𝑡 ↔ ∀𝑡 ∈ 𝑧 ¬ 𝑡 ∈ 𝑡))
62	59, 60, 61	3anbi123d 1437	. . . . . . . . . . . . . . . 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 3679	. . . . . . . . . . . . . 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 248	. . . . . . . . . . . . 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 7826	. . . . . . . . . . . . 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 1454	. . . . . . . . . . 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 4936	. . . . . . . . . . 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 6478	. . . . . . . . . . 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	biimtrrid 243	. . . . 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 149	. . 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 470	. . . 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 5266	. . . . 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 3322	. . . . 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 632	. . . 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 233	. . 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 412	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥((𝑥 ⊊ 𝐴 ∧ Tr 𝑥) → 𝑥 ∈ 𝐴) → (Tr 𝐴 ∧ ∀𝑧 ∈ 𝐴 ¬ 𝑧 ∈ 𝑧)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1537   = wceq 1539   ∈ wcel 2107  {cab 2713  ∀wral 3060  ∃wrex 3069  Vcvv 3479   ∪ cun 3948   ⊆ wss 3950   ⊊ wpss 3951  {csn 4625  ∪ cuni 4906  Tr wtr 5258  suc csuc 6385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-pw 4601  df-sn 4626  df-pr 4628  df-uni 4907  df-iun 4992  df-tr 5259  df-suc 6389

This theorem is referenced by:  dfon2lem4  35788  dfon2lem5  35789  dfon2lem7  35791  dfon2lem8  35792  dfon2lem9  35793  dfon2  35794