Theorem wfac8prim 45358

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 45347. 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 45315	. . 3 ⊢ Tr ∪ (𝑅1 “ On)
2		wfax.1	. . . 4 ⊢ 𝑊 = ∪ (𝑅1 “ On)
3		treq 5214	. . . 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 10414	. . . . 5 ⊢ ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑡∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡))
7		uniwf 9743	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ ∪ 𝑥 ∈ ∪ (𝑅1 “ On))
8		inss2 4192	. . . . . . . . . . 11 ⊢ (𝑡 ∩ ∪ 𝑥) ⊆ ∪ 𝑥
9		sswf 9732	. . . . . . . . . . 11 ⊢ ((∪ 𝑥 ∈ ∪ (𝑅1 “ On) ∧ (𝑡 ∩ ∪ 𝑥) ⊆ ∪ 𝑥) → (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
10	8, 9	mpan2 692	. . . . . . . . . 10 ⊢ (∪ 𝑥 ∈ ∪ (𝑅1 “ On) → (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
11	7, 10	sylbi 217	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
12	2	eleq2i 2829	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∈ ∪ (𝑅1 “ On))
13	2	eleq2i 2829	. . . . . . . . 9 ⊢ ((𝑡 ∩ ∪ 𝑥) ∈ 𝑊 ↔ (𝑡 ∩ ∪ 𝑥) ∈ ∪ (𝑅1 “ On))
14	11, 12, 13	3imtr4i 292	. . . . . . . 8 ⊢ (𝑥 ∈ 𝑊 → (𝑡 ∩ ∪ 𝑥) ∈ 𝑊)
15		inss1 4191	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∩ 𝑡) ⊆ 𝑧
16		elssuni 4896	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑥 → 𝑧 ⊆ ∪ 𝑥)
17	15, 16	sstrid 3947	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) ⊆ ∪ 𝑥)
18		dfss 3922	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∩ 𝑡) ⊆ ∪ 𝑥 ↔ (𝑧 ∩ 𝑡) = ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥))
19	17, 18	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) = ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥))
20		inass 4182	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∩ 𝑡) ∩ ∪ 𝑥) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))
21	19, 20	eqtrdi 2788	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝑥 → (𝑧 ∩ 𝑡) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
22	21	eleq2d 2823	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑥 → (𝑣 ∈ (𝑧 ∩ 𝑡) ↔ 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
23	22	eubidv 2587	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝑥 → (∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) ↔ ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
24	23	ralbiia 3082	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡) ↔ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
25		ineq2 4168	. . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (𝑧 ∩ 𝑦) = (𝑧 ∩ (𝑡 ∩ ∪ 𝑥)))
26	25	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑡 ∩ ∪ 𝑥) → (𝑣 ∈ (𝑧 ∩ 𝑦) ↔ 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))))
27	26	eubidv 2587	. . . . . . . . . . 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 3578	. . . . . . . . 9 ⊢ (((𝑡 ∩ ∪ 𝑥) ∈ 𝑊 ∧ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ (𝑡 ∩ ∪ 𝑥))) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
30	24, 29	sylan2b 595	. . . . . . . 8 ⊢ (((𝑡 ∩ ∪ 𝑥) ∈ 𝑊 ∧ ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑡)) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
31	14, 30	sylan 581	. . . . . . 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 1935	. . . . 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 3054	. . 3 ⊢ ∀𝑥 ∈ 𝑊 ((∀𝑧 ∈ 𝑥 𝑧 ≠ ∅ ∧ ∀𝑧 ∈ 𝑥 ∀𝑤 ∈ 𝑥 (𝑧 ≠ 𝑤 → (𝑧 ∩ 𝑤) = ∅)) → ∃𝑦 ∈ 𝑊 ∀𝑧 ∈ 𝑥 ∃!𝑣 𝑣 ∈ (𝑧 ∩ 𝑦))
36		modelac8prim 45348	. . 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 1542  ∃wex 1781   ∈ wcel 2114  ∃!weu 2569   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  ∪ cuni 4865  Tr wtr 5207   “ cima 5635  Oncon0 6325  𝑅₁cr1 9686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-om 7819  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-r1 9688  df-rank 9689  df-ac 10038

This theorem is referenced by: (None)