Theorem tz9.1regs 35271

Description: Every set has a transitive closure (the smallest transitive extension). This version of tz9.1 9642 depends on ax-regs 35263 instead of ax-reg 9501 and ax-inf2 9554. This suggests a possible answer to the third question posed in tz9.1 9642, 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 9501 applies to them directly, but in a finitist context it seems that an axiom like ax-regs 35263 is required since countably infinite classes are proper classes.

A related candidate for the missing property is the non-existence of infinite descending ∈-chains, proven as noinfep 9573 using ax-reg 9501 and ax-inf2 9554 and as noinfepregs 35270 using ax-regs 35263. If all sets are finite, then the existence of such a chain implies there is a set which does not have a transitive closure, as shown in fineqvinfep 35262. (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 3960	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
3		cleq1lem 14909	. . . . . 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 1922	. . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦) ↔ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
6	2, 5	3anbi13d 1441	. . 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 1923	. 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 3960	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑥))
9		cleq1lem 14909	. . . . . . 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 1922	. . . . 5 ⊢ (𝑧 = 𝑤 → (∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦) ↔ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
12	8, 11	3anbi13d 1441	. . . 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 1923	. . 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 3445	. . . . 5 ⊢ 𝑧 ∈ V
15		3simpa 1149	. . . . . . . . 9 ⊢ ((𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) → (𝑤 ⊆ 𝑥 ∧ Tr 𝑥))
16	15	eximi 1837	. . . . . . . 8 ⊢ (∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) → ∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥))
17		intexab 5292	. . . . . . . 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 3074	. . . . . 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 7909	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ ∀𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) → ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
21	14, 19, 20	sylancr 588	. . . . 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 7690	. . . . 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 588	. . . 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 4131	. . . . 5 ⊢ 𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
25		uniun 4887	. . . . . . 7 ⊢ ∪ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) = (∪ 𝑧 ∪ ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
26		uniiun 5015	. . . . . . . . . 10 ⊢ ∪ 𝑧 = ∪ 𝑤 ∈ 𝑧 𝑤
27		ssmin 4923	. . . . . . . . . . . 12 ⊢ 𝑤 ⊆ ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
28	27	rgenw 3056	. . . . . . . . . . 11 ⊢ ∀𝑤 ∈ 𝑧 𝑤 ⊆ ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
29		ss2iun 4966	. . . . . . . . . . 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 3981	. . . . . . . . 9 ⊢ ∪ 𝑧 ⊆ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
32		ssun4 4134	. . . . . . . . 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 5223	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
35		sseq2 3961	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑤 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑦))
36		treq 5213	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
37	35, 36	anbi12d 633	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((𝑤 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)))
38	37	cbvabv 2807	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑦 ∣ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)}
39	38	eqabri 2879	. . . . . . . . . . . . . 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 3058	. . . . . . . . . . . 12 ⊢ Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
42	41	rgenw 3056	. . . . . . . . . . 11 ⊢ ∀𝑤 ∈ 𝑧 Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
43		triun 5220	. . . . . . . . . . 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 5207	. . . . . . . . . 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 4134	. . . . . . . . 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 4144	. . . . . . 7 ⊢ (∪ 𝑧 ∪ ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
50	25, 49	eqsstri 3981	. . . . . 6 ⊢ ∪ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
51		df-tr 5207	. . . . . 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 3928	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝑦 → (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))
54		trss 5216	. . . . . . . . . . . 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 3472	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
59	58	inteqi 4907	. . . . . . . . . . 11 ⊢ ∩ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} = ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
60		vex 3445	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
61	37	intminss 4930	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ V ∧ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)) → ∩ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
62	60, 61	mpan 691	. . . . . . . . . . 11 ⊢ ((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → ∩ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
63	59, 62	eqsstrrid 3974	. . . . . . . . . 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 3128	. . . . . . . 8 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
66		iunss 5001	. . . . . . . 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 4143	. . . . . . . 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 592	. . . . . 6 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)
71	70	ax-gen 1797	. . . . 5 ⊢ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)
72	24, 52, 71	3pm3.2i 1341	. . . 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 3961	. . . . . . 7 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → (𝑧 ⊆ 𝑢 ↔ 𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})))
74		treq 5213	. . . . . . 7 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → (Tr 𝑢 ↔ Tr (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})))
75		sseq1 3960	. . . . . . . . 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 1922	. . . . . . 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 1439	. . . . . 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 3552	. . . . 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 3961	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑧 ⊆ 𝑢 ↔ 𝑧 ⊆ 𝑥))
81		treq 5213	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (Tr 𝑢 ↔ Tr 𝑥))
82		sseq1 3960	. . . . . . . . 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 1922	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑢 ⊆ 𝑦) ↔ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
85	80, 81, 84	3anbi123d 1439	. . . . . 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 2039	. . . . 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 35268	. 2 ⊢ ∃𝑥(𝑧 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
90	1, 7, 89	vtocl 3516	1 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  ∀wral 3052  {crab 3400  Vcvv 3441   ∪ cun 3900   ⊆ wss 3902  ∪ cuni 4864  ∩ cint 4903  ∪ ciun 4947  Tr wtr 5206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682  ax-regs 35263

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-sn 4582  df-pr 4584  df-uni 4865  df-int 4904  df-iun 4949  df-iin 4950  df-tr 5207

This theorem is referenced by: (None)