Theorem wfac8prim 44965

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 44954. 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 44922	. . 3 ⊢ Tr ∪ (𝑅1 “ On)
2		wfax.1	. . . 4 ⊢ 𝑊 = ∪ (𝑅1 “ On)
3		treq 5217	. . . 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 10421	. . . . 5 ⊢ ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑡∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡))
7		uniwf 9748	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝑥 ∈ ∪ (𝑅1 “ On))
8		inss2 4197	. . . . . . . . . . 11 ⊢ (𝑡 ∩ ∪ 𝑥) ⊆ ∪ 𝑥
9		sswf 9737	. . . . . . . . . . 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 2820	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
13	2	eleq2i 2820	. . . . . . . . 9 ⊢ ((𝑡 ∩ ∪ 𝑥) ∈ 𝑊 ↔ (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
14	11, 12, 13	3imtr4i 292	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑊 → (𝑡 ∩ ∪ 𝑥) ∈ 𝑊)
15		inss1 4196	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∩ 𝑡) ⊆ 𝑧
16		elssuni 4897	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑥 → 𝑧 ⊆ ∪ 𝑥)
17	15, 16	sstrid 3955	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) ⊆ ∪ 𝑥)
18		dfss 3930	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∩ 𝑡) ⊆ ∪ 𝑥 ↔ (𝑧 ∩ 𝑡) = ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥))
19	17, 18	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) = ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥))
20		inass 4187	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))
21	19, 20	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
22	21	eleq2d 2814	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → (𝑣 ∈ (𝑧 ∩ 𝑡) ↔ 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
23	22	eubidv 2579	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
24	23	ralbiia 3073	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
25		ineq2 4173	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (𝑧 ∩ 𝑦) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
26	25	eleq2d 2814	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
27	26	eubidv 2579	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
28	27	ralbidv 3156	. . . . . . . . . 10 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
29	28	rspcev 3585	. . . . . . . . 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 1933	. . . . 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 3046	. . 3 ⊢ ∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
36		modelac8prim 44955	. . 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 1540  ∃wex 1779   ∈ wcel 2109  ∃!weu 2561   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  ∪ cuni 4867  Tr wtr 5209   “ cima 5634  Oncon0 6320  𝑅₁cr1 9691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-ac2 10392

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-r1 9693  df-rank 9694  df-ac 10045

This theorem is referenced by: (None)