Theorem tz9.1regs 35266

Description: Every set has a transitive closure (the smallest transitive extension). This version of tz9.1 9639 depends on ax-regs 35258 instead of ax-reg 9496 and ax-inf2 9551. This suggests a possible answer to the third question posed in tz9.1 9639, 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 9496 applies to them directly, but in a finitist context it seems that an axiom like ax-regs 35258 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 9570 using ax-reg 9496 and ax-inf2 9551 and as noinfepregs 35265 using ax-regs 35258. 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 35257. (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 3942	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
3		cleq1lem 14933	. . . . . 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 3942	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑥))
9		cleq1lem 14933	. . . . . . 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 3431	. . . . 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 5276	. . . . . . . 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 3072	. . . . . 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 7905	. . . . . 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 7686	. . . . 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 4109	. . . . 5 ⊢ 𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
25		uniun 4863	. . . . . . 7 ⊢ ∪ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) = (∪ 𝑧 ∪ ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
26		uniiun 4990	. . . . . . . . . 10 ⊢ ∪ 𝑧 = ∪ 𝑤 ∈ 𝑧 𝑤
27		ssmin 4899	. . . . . . . . . . . 12 ⊢ 𝑤 ⊆ ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
28	27	rgenw 3053	. . . . . . . . . . 11 ⊢ ∀𝑤 ∈ 𝑧 𝑤 ⊆ ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
29		ss2iun 4942	. . . . . . . . . . 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 3963	. . . . . . . . 9 ⊢ ∪ 𝑧 ⊆ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
32		ssun4 4112	. . . . . . . . 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 5199	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
35		sseq2 3943	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑤 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑦))
36		treq 5188	. . . . . . . . . . . . . . . . 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 2805	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑦 ∣ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)}
39	38	eqabri 2877	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦))
40	39	simprbi 497	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} → Tr 𝑦)
41	34, 40	mprg 3055	. . . . . . . . . . . 12 ⊢ Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
42	41	rgenw 3053	. . . . . . . . . . 11 ⊢ ∀𝑤 ∈ 𝑧 Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
43		triun 5196	. . . . . . . . . . 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 5182	. . . . . . . . . 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 4112	. . . . . . . . 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 4122	. . . . . . 7 ⊢ (∪ 𝑧 ∪ ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
50	25, 49	eqsstri 3963	. . . . . 6 ⊢ ∪ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
51		df-tr 5182	. . . . . 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 3911	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝑦 → (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))
54		trss 5191	. . . . . . . . . . . 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 3458	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
59	58	inteqi 4883	. . . . . . . . . . 11 ⊢ ∩ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} = ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
60		vex 3431	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
61	37	intminss 4906	. . . . . . . . . . . 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 3956	. . . . . . . . . 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 3126	. . . . . . . 8 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
66		iunss 4976	. . . . . . . 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 4121	. . . . . . . 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 3943	. . . . . . 7 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → (𝑧 ⊆ 𝑢 ↔ 𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})))
74		treq 5188	. . . . . . 7 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → (Tr 𝑢 ↔ Tr (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})))
75		sseq1 3942	. . . . . . . . 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 3537	. . . . 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 3943	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑧 ⊆ 𝑢 ↔ 𝑧 ⊆ 𝑥))
81		treq 5188	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (Tr 𝑢 ↔ Tr 𝑥))
82		sseq1 3942	. . . . . . . . 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 35263	. 2 ⊢ ∃𝑥(𝑧 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
90	1, 7, 89	vtocl 3501	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 2713  ∀wral 3049  {crab 3387  Vcvv 3427   ∪ cun 3883   ⊆ wss 3885  ∪ cuni 4840  ∩ cint 4879  ∪ ciun 4923  Tr wtr 5181

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 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-pr 5364  ax-un 7678  ax-regs 35258

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 2538  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rab 3388  df-v 3429  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-sn 4558  df-pr 4560  df-uni 4841  df-int 4880  df-iun 4925  df-iin 4926  df-tr 5182

This theorem is referenced by: (None)