Theorem wfac8prim 45833

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 45822. 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 45790	. . 3 ⊢ Tr ∪ (𝑅1 “ On)
2		wfax.1	. . . 4 ⊢ 𝑊 = ∪ (𝑅1 “ On)
3		treq 5223	. . . 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 234	. 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 9805	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝑥 ∈ ∪ (𝑅1 “ On))
8		inss2 4186	. . . . . . . . . . 11 ⊢ (𝑡 ∩ ∪ 𝑥) ⊆ ∪ 𝑥
9		sswf 9794	. . . . . . . . . . 11 ⊢ ((∪ 𝑥 ∈ ∪ (𝑅1 “ On) ∧ (𝑡 ∩ ∪ 𝑥) ⊆ ∪ 𝑥) → (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
10	8, 9	mpan2 704	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ ∪ (𝑅1 “ On) → (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
11	7, 10	sylbi 220	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
12	2	eleq2i 2854	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
13	2	eleq2i 2854	. . . . . . . . 9 ⊢ ((𝑡 ∩ ∪ 𝑥) ∈ 𝑊 ↔ (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
14	11, 12, 13	3imtr4i 295	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑊 → (𝑡 ∩ ∪ 𝑥) ∈ 𝑊)
15		inss1 4185	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∩ 𝑡) ⊆ 𝑧
16		elssuni 4902	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑥 → 𝑧 ⊆ ∪ 𝑥)
17	15, 16	sstrid 3945	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) ⊆ ∪ 𝑥)
18		dfss 3921	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∩ 𝑡) ⊆ ∪ 𝑥 ↔ (𝑧 ∩ 𝑡) = ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥))
19	17, 18	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) = ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥))
20		inass 4176	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))
21	19, 20	eqtrdi 2813	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
22	21	eleq2d 2848	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → (𝑣 ∈ (𝑧 ∩ 𝑡) ↔ 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
23	22	eubidv 2613	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
24	23	ralbiia 3108	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
25		ineq2 4163	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (𝑧 ∩ 𝑦) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
26	25	eleq2d 2848	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
27	26	eubidv 2613	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
28	27	ralbidv 3187	. . . . . . . . . 10 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
29	28	rspcev 3579	. . . . . . . . 9 ⊢ (((𝑡 ∩ ∪ 𝑥) ∈ 𝑊 ∧ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
30	24, 29	sylan2b 606	. . . . . . . 8 ⊢ (((𝑡 ∩ ∪ 𝑥) ∈ 𝑊 ∧ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡)) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
31	14, 30	sylan 592	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑊 ∧ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡)) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
32	31	ex 418	. . . . . 6 ⊢ (𝑥 ∈ 𝑊 → (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
33	32	exlimdv 1966	. . . . 5 ⊢ (𝑥 ∈ 𝑊 → (∃𝑡∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦)))
34	6, 33	syl5 35	. . . 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 3080	. . 3 ⊢ ∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
36		modelac8prim 45823	. . 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 236	. 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)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃!weu 2595   ≠ wne 2957  ∀wral 3078  ∃wrex 3088   ∩ cin 3901   ⊆ wss 3902  ∅c0 4282  ∪ cuni 4870  Tr wtr 5216   “ cima 5662  Oncon0 6361  𝑅₁cr1 9748

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-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-ac2 10469

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-om 7867  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-r1 9750  df-rank 9751  df-ac 10123

This theorem is used by: (None)