Theorem wfac8prim 45035

Description: The class of well-founded sets 𝑊 models the Axiom of Choice. Since the previous theorems show that all the ZF axioms hold in 𝑊, we may use any statement that ZF proves is equivalent to Choice to prove this. We use ac8prim 45024. Part of Corollary II.2.12 of [Kunen2] p. 114. (Contributed by Eric Schmidt, 19-Oct-2025.)

Hypothesis

Ref	Expression
wfax.1	⊢ 𝑊 = ∪ (𝑅1 “ On)

Assertion

Ref	Expression
wfac8prim	⊢ ∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑊 ∀𝑤 ∈ 𝑊 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑊 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 ∀𝑣 ∈ 𝑊 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))

Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑊

Proof of Theorem wfac8prim

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trwf 44992	. . 3 ⊢ Tr ∪ (𝑅1 “ On)
2		wfax.1	. . . 4 ⊢ 𝑊 = ∪ (𝑅1 “ On)
3		treq 5200	. . . 4 ⊢ (𝑊 = ∪ (𝑅1 “ On) → (Tr 𝑊 ↔ Tr ∪ (𝑅1 “ On)))
4	2, 3	ax-mp 5	. . 3 ⊢ (Tr 𝑊 ↔ Tr ∪ (𝑅1 “ On))
5	1, 4	mpbir 231	. 2 ⊢ Tr 𝑊
6		ac8 10378	. . . . 5 ⊢ ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑡∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡))
7		uniwf 9707	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝑥 ∈ ∪ (𝑅1 “ On))
8		inss2 4183	. . . . . . . . . . 11 ⊢ (𝑡 ∩ ∪ 𝑥) ⊆ ∪ 𝑥
9		sswf 9696	. . . . . . . . . . 11 ⊢ ((∪ 𝑥 ∈ ∪ (𝑅1 “ On) ∧ (𝑡 ∩ ∪ 𝑥) ⊆ ∪ 𝑥) → (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
10	8, 9	mpan2 691	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ ∪ (𝑅1 “ On) → (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
11	7, 10	sylbi 217	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
12	2	eleq2i 2823	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
13	2	eleq2i 2823	. . . . . . . . 9 ⊢ ((𝑡 ∩ ∪ 𝑥) ∈ 𝑊 ↔ (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
14	11, 12, 13	3imtr4i 292	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑊 → (𝑡 ∩ ∪ 𝑥) ∈ 𝑊)
15		inss1 4182	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∩ 𝑡) ⊆ 𝑧
16		elssuni 4884	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑥 → 𝑧 ⊆ ∪ 𝑥)
17	15, 16	sstrid 3941	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) ⊆ ∪ 𝑥)
18		dfss 3916	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∩ 𝑡) ⊆ ∪ 𝑥 ↔ (𝑧 ∩ 𝑡) = ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥))
19	17, 18	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) = ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥))
20		inass 4173	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))
21	19, 20	eqtrdi 2782	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
22	21	eleq2d 2817	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → (𝑣 ∈ (𝑧 ∩ 𝑡) ↔ 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
23	22	eubidv 2581	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
24	23	ralbiia 3076	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
25		ineq2 4159	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (𝑧 ∩ 𝑦) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
26	25	eleq2d 2817	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
27	26	eubidv 2581	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
28	27	ralbidv 3155	. . . . . . . . . 10 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
29	28	rspcev 3572	. . . . . . . . 9 ⊢ (((𝑡 ∩ ∪ 𝑥) ∈ 𝑊 ∧ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
30	24, 29	sylan2b 594	. . . . . . . 8 ⊢ (((𝑡 ∩ ∪ 𝑥) ∈ 𝑊 ∧ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡)) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
31	14, 30	sylan 580	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑊 ∧ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡)) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
32	31	ex 412	. . . . . 6 ⊢ (𝑥 ∈ 𝑊 → (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
33	32	exlimdv 1934	. . . . 5 ⊢ (𝑥 ∈ 𝑊 → (∃𝑡∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
34	6, 33	syl5 34	. . . 4 ⊢ (𝑥 ∈ 𝑊 → ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
35	34	rgen 3049	. . 3 ⊢ ∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
36		modelac8prim 45025	. . 3 ⊢ (Tr 𝑊 → (∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)) ↔ ∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑊 ∀𝑤 ∈ 𝑊 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑊 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 ∀𝑣 ∈ 𝑊 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))))
37	35, 36	mpbii 233	. 2 ⊢ (Tr 𝑊 → ∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑊 ∀𝑤 ∈ 𝑊 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑊 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 ∀𝑣 ∈ 𝑊 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤))))
38	5, 37	ax-mp 5	1 ⊢ ∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 𝑤 ∈ 𝑧) ∧ ∀𝑧 ∈ 𝑊 ∀𝑤 ∈ 𝑊 ((𝑧 ∈ 𝑥 ∧ 𝑤 ∈ 𝑥) → (¬ 𝑧 = 𝑤 → ∀𝑦 ∈ 𝑊 (𝑦 ∈ 𝑧 → ¬ 𝑦 ∈ 𝑤)))) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑊 (𝑧 ∈ 𝑥 → ∃𝑤 ∈ 𝑊 ∀𝑣 ∈ 𝑊 ((𝑣 ∈ 𝑧 ∧ 𝑣 ∈ 𝑦) ↔ 𝑣 = 𝑤)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃!weu 2563   ≠ wne 2928  ∀wral 3047  ∃wrex 3056   ∩ cin 3896   ⊆ wss 3897  ∅c0 4278  ∪ cuni 4854  Tr wtr 5193   “ cima 5614  Oncon0 6301  𝑅₁cr1 9650

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663  ax-ac2 10349

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-r1 9652  df-rank 9653  df-ac 10002

This theorem is referenced by: (None)