Theorem tz9.1regs 35151

Description: Every set has a transitive closure (the smallest transitive extension). This version of tz9.1 9626 depends on ax-regs 35145 instead of ax-reg 9485 and ax-inf2 9538. This suggests a possible answer to the third question posed in tz9.1 9626, namely that the missing property is that countably infinite classes must obey regularity. In ZF set theory we can prove this by showing that countably infinite classes are sets and thus ax-reg 9485 applies to them directly, but in a finitist context it seems that an axiom like ax-regs 35145 is required since countably infinite classes are proper classes. (Contributed by BTernaryTau, 31-Dec-2025.)

Hypothesis

Ref	Expression
tz9.1regs.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tz9.1regs	⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem tz9.1regs

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

Step	Hyp	Ref	Expression
1		tz9.1regs.1	. 2 ⊢ 𝐴 ∈ V
2		sseq1 3956	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
3		cleq1lem 14891	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) ↔ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)))
4	3	imbi1d 341	. . . . 5 ⊢ (𝑧 = 𝐴 → (((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦) ↔ ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
5	4	albidv 1921	. . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦) ↔ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
6	2, 5	3anbi13d 1440	. . 3 ⊢ (𝑧 = 𝐴 → ((𝑧 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) ↔ (𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))))
7	6	exbidv 1922	. 2 ⊢ (𝑧 = 𝐴 → (∃𝑥(𝑧 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) ↔ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))))
8		sseq1 3956	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑥))
9		cleq1lem 14891	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) ↔ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)))
10	9	imbi1d 341	. . . . . 6 ⊢ (𝑧 = 𝑤 → (((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦) ↔ ((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
11	10	albidv 1921	. . . . 5 ⊢ (𝑧 = 𝑤 → (∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦) ↔ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
12	8, 11	3anbi13d 1440	. . . 4 ⊢ (𝑧 = 𝑤 → ((𝑧 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) ↔ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))))
13	12	exbidv 1922	. . 3 ⊢ (𝑧 = 𝑤 → (∃𝑥(𝑧 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) ↔ ∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))))
14		vex 3441	. . . . 5 ⊢ 𝑧 ∈ V
15		3simpa 1148	. . . . . . . . 9 ⊢ ((𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) → (𝑤 ⊆ 𝑥 ∧ Tr 𝑥))
16	15	eximi 1836	. . . . . . . 8 ⊢ (∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) → ∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥))
17		intexab 5286	. . . . . . . 8 ⊢ (∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥) ↔ ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
18	16, 17	sylib 218	. . . . . . 7 ⊢ (∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) → ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
19	18	ralimi 3070	. . . . . 6 ⊢ (∀𝑤 ∈ 𝑧 ∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) → ∀𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
20		iunexg 7901	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ ∀𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) → ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
21	14, 19, 20	sylancr 587	. . . . 5 ⊢ (∀𝑤 ∈ 𝑧 ∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) → ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
22		unexg 7682	. . . . 5 ⊢ ((𝑧 ∈ V ∧ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ∈ V)
23	14, 21, 22	sylancr 587	. . . 4 ⊢ (∀𝑤 ∈ 𝑧 ∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ∈ V)
24		ssun1 4127	. . . . 5 ⊢ 𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
25		uniun 4881	. . . . . . 7 ⊢ ∪ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) = (∪ 𝑧 ∪ ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
26		uniiun 5009	. . . . . . . . . 10 ⊢ ∪ 𝑧 = ∪ 𝑤 ∈ 𝑧 𝑤
27		ssmin 4917	. . . . . . . . . . . 12 ⊢ 𝑤 ⊆ ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
28	27	rgenw 3052	. . . . . . . . . . 11 ⊢ ∀𝑤 ∈ 𝑧 𝑤 ⊆ ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
29		ss2iun 4960	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ 𝑧 𝑤 ⊆ ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} → ∪ 𝑤 ∈ 𝑧 𝑤 ⊆ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
30	28, 29	ax-mp 5	. . . . . . . . . 10 ⊢ ∪ 𝑤 ∈ 𝑧 𝑤 ⊆ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
31	26, 30	eqsstri 3977	. . . . . . . . 9 ⊢ ∪ 𝑧 ⊆ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
32		ssun4 4130	. . . . . . . . 9 ⊢ (∪ 𝑧 ⊆ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} → ∪ 𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}))
33	31, 32	ax-mp 5	. . . . . . . 8 ⊢ ∪ 𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
34		trint 5217	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
35		sseq2 3957	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑤 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑦))
36		treq 5207	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
37	35, 36	anbi12d 632	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((𝑤 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)))
38	37	cbvabv 2803	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑦 ∣ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)}
39	38	eqabri 2875	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦))
40	39	simprbi 496	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} → Tr 𝑦)
41	34, 40	mprg 3054	. . . . . . . . . . . 12 ⊢ Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
42	41	rgenw 3052	. . . . . . . . . . 11 ⊢ ∀𝑤 ∈ 𝑧 Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
43		triun 5214	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ 𝑧 Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} → Tr ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
44	42, 43	ax-mp 5	. . . . . . . . . 10 ⊢ Tr ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
45		df-tr 5201	. . . . . . . . . 10 ⊢ (Tr ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
46	44, 45	mpbi 230	. . . . . . . . 9 ⊢ ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
47		ssun4 4130	. . . . . . . . 9 ⊢ (∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} → ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}))
48	46, 47	ax-mp 5	. . . . . . . 8 ⊢ ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
49	33, 48	unssi 4140	. . . . . . 7 ⊢ (∪ 𝑧 ∪ ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
50	25, 49	eqsstri 3977	. . . . . 6 ⊢ ∪ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
51		df-tr 5201	. . . . . 6 ⊢ (Tr (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ↔ ∪ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}))
52	50, 51	mpbir 231	. . . . 5 ⊢ Tr (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
53		ssel 3924	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝑦 → (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))
54		trss 5210	. . . . . . . . . . . 12 ⊢ (Tr 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ⊆ 𝑦))
55	53, 54	sylan9 507	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑤 ∈ 𝑧 → 𝑤 ⊆ 𝑦))
56		simpr 484	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦)
57	55, 56	jctird 526	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑤 ∈ 𝑧 → (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)))
58		rabab 3468	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
59	58	inteqi 4901	. . . . . . . . . . 11 ⊢ ∩ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} = ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
60		vex 3441	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
61	37	intminss 4924	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ V ∧ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)) → ∩ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
62	60, 61	mpan 690	. . . . . . . . . . 11 ⊢ ((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → ∩ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
63	59, 62	eqsstrrid 3970	. . . . . . . . . 10 ⊢ ((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
64	57, 63	syl6 35	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑤 ∈ 𝑧 → ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦))
65	64	ralrimiv 3124	. . . . . . . 8 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
66		iunss 4995	. . . . . . . 8 ⊢ (∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦 ↔ ∀𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
67	65, 66	sylibr 234	. . . . . . 7 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
68		unss 4139	. . . . . . . 8 ⊢ ((𝑧 ⊆ 𝑦 ∧ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦) ↔ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)
69	68	biimpi 216	. . . . . . 7 ⊢ ((𝑧 ⊆ 𝑦 ∧ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)
70	67, 69	syldan 591	. . . . . 6 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)
71	70	ax-gen 1796	. . . . 5 ⊢ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)
72	24, 52, 71	3pm3.2i 1340	. . . 4 ⊢ (𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ∧ Tr (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦))
73		sseq2 3957	. . . . . . 7 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → (𝑧 ⊆ 𝑢 ↔ 𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})))
74		treq 5207	. . . . . . 7 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → (Tr 𝑢 ↔ Tr (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})))
75		sseq1 3956	. . . . . . . . 9 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → (𝑢 ⊆ 𝑦 ↔ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦))
76	75	imbi2d 340	. . . . . . . 8 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → (((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑢 ⊆ 𝑦) ↔ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)))
77	76	albidv 1921	. . . . . . 7 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → (∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑢 ⊆ 𝑦) ↔ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)))
78	73, 74, 77	3anbi123d 1438	. . . . . 6 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → ((𝑧 ⊆ 𝑢 ∧ Tr 𝑢 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑢 ⊆ 𝑦)) ↔ (𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ∧ Tr (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦))))
79	78	spcegv 3548	. . . . 5 ⊢ ((𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ∈ V → ((𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ∧ Tr (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)) → ∃𝑢(𝑧 ⊆ 𝑢 ∧ Tr 𝑢 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑢 ⊆ 𝑦))))
80		sseq2 3957	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑧 ⊆ 𝑢 ↔ 𝑧 ⊆ 𝑥))
81		treq 5207	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (Tr 𝑢 ↔ Tr 𝑥))
82		sseq1 3956	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (𝑢 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑦))
83	82	imbi2d 340	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → (((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑢 ⊆ 𝑦) ↔ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
84	83	albidv 1921	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑢 ⊆ 𝑦) ↔ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
85	80, 81, 84	3anbi123d 1438	. . . . . 6 ⊢ (𝑢 = 𝑥 → ((𝑧 ⊆ 𝑢 ∧ Tr 𝑢 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑢 ⊆ 𝑦)) ↔ (𝑧 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))))
86	85	cbvexvw 2038	. . . . 5 ⊢ (∃𝑢(𝑧 ⊆ 𝑢 ∧ Tr 𝑢 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑢 ⊆ 𝑦)) ↔ ∃𝑥(𝑧 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
87	79, 86	imbitrdi 251	. . . 4 ⊢ ((𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ∈ V → ((𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ∧ Tr (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)) → ∃𝑥(𝑧 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))))
88	23, 72, 87	mpisyl 21	. . 3 ⊢ (∀𝑤 ∈ 𝑧 ∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) → ∃𝑥(𝑧 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
89	13, 88	setinds2regs 35150	. 2 ⊢ ∃𝑥(𝑧 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
90	1, 7, 89	vtocl 3512	1 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2711  ∀wral 3048  {crab 3396  Vcvv 3437   ∪ cun 3896   ⊆ wss 3898  ∪ cuni 4858  ∩ cint 4897  ∪ ciun 4941  Tr wtr 5200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372  ax-un 7674  ax-regs 35145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-sn 4576  df-pr 4578  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-tr 5201

This theorem is referenced by: (None)