Theorem wfac8prim 44954

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 44943. 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 44911	. . 3 ⊢ Tr ∪ (𝑅1 “ On)
2		wfax.1	. . . 4 ⊢ 𝑊 = ∪ (𝑅1 “ On)
3		treq 5234	. . . 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 10498	. . . . 5 ⊢ ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑡∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡))
7		uniwf 9825	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝑥 ∈ ∪ (𝑅1 “ On))
8		inss2 4211	. . . . . . . . . . 11 ⊢ (𝑡 ∩ ∪ 𝑥) ⊆ ∪ 𝑥
9		sswf 9814	. . . . . . . . . . 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 2825	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
13	2	eleq2i 2825	. . . . . . . . 9 ⊢ ((𝑡 ∩ ∪ 𝑥) ∈ 𝑊 ↔ (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
14	11, 12, 13	3imtr4i 292	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑊 → (𝑡 ∩ ∪ 𝑥) ∈ 𝑊)
15		inss1 4210	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∩ 𝑡) ⊆ 𝑧
16		elssuni 4910	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑥 → 𝑧 ⊆ ∪ 𝑥)
17	15, 16	sstrid 3968	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) ⊆ ∪ 𝑥)
18		dfss 3943	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∩ 𝑡) ⊆ ∪ 𝑥 ↔ (𝑧 ∩ 𝑡) = ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥))
19	17, 18	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) = ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥))
20		inass 4201	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))
21	19, 20	eqtrdi 2785	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
22	21	eleq2d 2819	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → (𝑣 ∈ (𝑧 ∩ 𝑡) ↔ 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
23	22	eubidv 2584	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
24	23	ralbiia 3079	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
25		ineq2 4187	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (𝑧 ∩ 𝑦) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
26	25	eleq2d 2819	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
27	26	eubidv 2584	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
28	27	ralbidv 3161	. . . . . . . . . 10 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
29	28	rspcev 3599	. . . . . . . . 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 1932	. . . . 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 3052	. . 3 ⊢ ∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
36		modelac8prim 44944	. . 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 1539  ∃wex 1778   ∈ wcel 2107  ∃!weu 2566   ≠ wne 2931  ∀wral 3050  ∃wrex 3059   ∩ cin 3923   ⊆ wss 3924  ∅c0 4306  ∪ cuni 4880  Tr wtr 5226   “ cima 5654  Oncon0 6349  𝑅₁cr1 9768

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723  ax-ac2 10469

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-pss 3944  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-int 4920  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-tr 5227  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6287  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7402  df-om 7856  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8379  df-rdg 8418  df-r1 9770  df-rank 9771  df-ac 10122

This theorem is referenced by: (None)