Theorem tz9.1regs 35647

Description: Every set has a transitive closure (the smallest transitive extension). This version of tz9.1 9711 depends on ax-regs 35639 instead of ax-reg 9567 and ax-inf2 9623. This suggests a possible answer to the third question posed in tz9.1 9711, 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 9567 applies to them directly, but in a finitist context it seems that an axiom like ax-regs 35639 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 9642 using ax-reg 9567 and ax-inf2 9623 and as noinfepregs 35646 using ax-regs 35639. 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 35638. (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 3959	. . . 4 ⊢ (𝑧 = 𝐴 → (𝑧 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑥))
3		cleq1lem 15057	. . . . . 6 ⊢ (𝑧 = 𝐴 → ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) ↔ (𝐴 ⊆ 𝑦 ∧ Tr 𝑦)))
4	3	imbi1d 344	. . . . 5 ⊢ (𝑧 = 𝐴 → (((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦) ↔ ((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
5	4	albidv 1953	. . . 4 ⊢ (𝑧 = 𝐴 → (∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦) ↔ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
6	2, 5	3anbi13d 1466	. . 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 1954	. 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 3959	. . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑥))
9		cleq1lem 15057	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) ↔ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)))
10	9	imbi1d 344	. . . . . 6 ⊢ (𝑧 = 𝑤 → (((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦) ↔ ((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
11	10	albidv 1953	. . . . 5 ⊢ (𝑧 = 𝑤 → (∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦) ↔ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
12	8, 11	3anbi13d 1466	. . . 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 1954	. . 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 3457	. . . . 5 ⊢ 𝑧 ∈ V
15		3simpa 1166	. . . . . . . . 9 ⊢ ((𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) → (𝑤 ⊆ 𝑥 ∧ Tr 𝑥))
16	15	eximi 1868	. . . . . . . 8 ⊢ (∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) → ∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥))
17		intexab 5314	. . . . . . . 8 ⊢ (∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥) ↔ ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
18	16, 17	sylib 221	. . . . . . 7 ⊢ (∃𝑥(𝑤 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)) → ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
19	18	ralimi 3101	. . . . . 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 7963	. . . . . 6 ⊢ ((𝑧 ∈ V ∧ ∀𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V) → ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ∈ V)
21	14, 19, 20	sylancr 599	. . . . 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 7748	. . . . 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 599	. . . 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 4893	. . . . . . 7 ⊢ ∪ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) = (∪ 𝑧 ∪ ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
26		uniiun 5021	. . . . . . . . . 10 ⊢ ∪ 𝑧 = ∪ 𝑤 ∈ 𝑧 𝑤
27		ssmin 4930	. . . . . . . . . . . 12 ⊢ 𝑤 ⊆ ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
28	27	rgenw 3082	. . . . . . . . . . 11 ⊢ ∀𝑤 ∈ 𝑧 𝑤 ⊆ ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
29		ss2iun 4973	. . . . . . . . . . 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 3980	. . . . . . . . 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 5234	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}Tr 𝑦 → Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
35		sseq2 3960	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (𝑤 ⊆ 𝑥 ↔ 𝑤 ⊆ 𝑦))
36		treq 5223	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
37	35, 36	anbi12d 644	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ((𝑤 ⊆ 𝑥 ∧ Tr 𝑥) ↔ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)))
38	37	cbvabv 2832	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑦 ∣ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)}
39	38	eqabri 2904	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ↔ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦))
40	39	simprbi 503	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} → Tr 𝑦)
41	34, 40	mprg 3084	. . . . . . . . . . . 12 ⊢ Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
42	41	rgenw 3082	. . . . . . . . . . 11 ⊢ ∀𝑤 ∈ 𝑧 Tr ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
43		triun 5231	. . . . . . . . . . 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 5217	. . . . . . . . . 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 233	. . . . . . . . 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 3980	. . . . . 6 ⊢ ∪ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
51		df-tr 5217	. . . . . 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 234	. . . . 5 ⊢ Tr (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})
53		ssel 3928	. . . . . . . . . . . 12 ⊢ (𝑧 ⊆ 𝑦 → (𝑤 ∈ 𝑧 → 𝑤 ∈ 𝑦))
54		trss 5226	. . . . . . . . . . . 12 ⊢ (Tr 𝑦 → (𝑤 ∈ 𝑦 → 𝑤 ⊆ 𝑦))
55	53, 54	sylan9 517	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑤 ∈ 𝑧 → 𝑤 ⊆ 𝑦))
56		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → Tr 𝑦)
57	55, 56	jctird 536	. . . . . . . . . 10 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑤 ∈ 𝑧 → (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)))
58		rabab 3483	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} = {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
59	58	inteqi 4914	. . . . . . . . . . 11 ⊢ ∩ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} = ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}
60		vex 3457	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
61	37	intminss 4937	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ V ∧ (𝑤 ⊆ 𝑦 ∧ Tr 𝑦)) → ∩ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
62	60, 61	mpan 703	. . . . . . . . . . 11 ⊢ ((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → ∩ {𝑥 ∈ V ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
63	59, 62	eqsstrrid 3973	. . . . . . . . . 10 ⊢ ((𝑤 ⊆ 𝑦 ∧ Tr 𝑦) → ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
64	57, 63	syl6 36	. . . . . . . . 9 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑤 ∈ 𝑧 → ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦))
65	64	ralrimiv 3155	. . . . . . . 8 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → ∀𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
66		iunss 5007	. . . . . . . 8 ⊢ (∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦 ↔ ∀𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦)
67	65, 66	sylibr 237	. . . . . . 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 219	. . . . . . 7 ⊢ ((𝑧 ⊆ 𝑦 ∧ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)} ⊆ 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)
70	67, 69	syldan 603	. . . . . 6 ⊢ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)
71	70	ax-gen 1828	. . . . 5 ⊢ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦)
72	24, 52, 71	3pm3.2i 1358	. . . 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 3960	. . . . . . 7 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → (𝑧 ⊆ 𝑢 ↔ 𝑧 ⊆ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})))
74		treq 5223	. . . . . . 7 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → (Tr 𝑢 ↔ Tr (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)})))
75		sseq1 3959	. . . . . . . . 9 ⊢ (𝑢 = (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) → (𝑢 ⊆ 𝑦 ↔ (𝑧 ∪ ∪ 𝑤 ∈ 𝑧 ∩ {𝑥 ∣ (𝑤 ⊆ 𝑥 ∧ Tr 𝑥)}) ⊆ 𝑦))
76	75	imbi2d 343	. . . . . . . 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 1953	. . . . . . 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 1464	. . . . . 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 3554	. . . . 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 3960	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑧 ⊆ 𝑢 ↔ 𝑧 ⊆ 𝑥))
81		treq 5223	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (Tr 𝑢 ↔ Tr 𝑥))
82		sseq1 3959	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (𝑢 ⊆ 𝑦 ↔ 𝑥 ⊆ 𝑦))
83	82	imbi2d 343	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → (((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑢 ⊆ 𝑦) ↔ ((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
84	83	albidv 1953	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑢 ⊆ 𝑦) ↔ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦)))
85	80, 81, 84	3anbi123d 1464	. . . . . 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 2070	. . . . 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 254	. . . 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 22	. . 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 35644	. 2 ⊢ ∃𝑥(𝑧 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝑧 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))
90	1, 7, 89	vtocl 3523	1 ⊢ ∃𝑥(𝐴 ⊆ 𝑥 ∧ Tr 𝑥 ∧ ∀𝑦((𝐴 ⊆ 𝑦 ∧ Tr 𝑦) → 𝑥 ⊆ 𝑦))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2740  ∀wral 3078  {crab 3414  Vcvv 3453   ∪ cun 3900   ⊆ wss 3902  ∪ cuni 4870  ∩ cint 4910  ∪ ciun 4954  Tr wtr 5216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-pr 5402  ax-un 7739  ax-regs 35639

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-sn 4588  df-pr 4590  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-tr 5217

This theorem is used by: (None)