Theorem omssaxinf2 44977

Description: A class that contains all ordinals up to and including ω models the Axiom of Infinity ax-inf2 9663. The antecedent of this theorem is not enough to guarantee that the class models the alternate axiom ax-inf 9660. (Contributed by Eric Schmidt, 19-Oct-2025.)

Assertion

Ref	Expression
omssaxinf2	⊢ ((ω ⊆ 𝑀 ∧ ω ∈ 𝑀) → ∃𝑥 ∈ 𝑀 (∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤   𝑥,𝑀,𝑦,𝑧

Allowed substitution hint:   𝑀(𝑤)

Proof of Theorem omssaxinf2

Step	Hyp	Ref	Expression
1		peano1 7892	. . . . 5 ⊢ ∅ ∈ ω
2		ssel 3957	. . . . 5 ⊢ (ω ⊆ 𝑀 → (∅ ∈ ω → ∅ ∈ 𝑀))
3	1, 2	mpi 20	. . . 4 ⊢ (ω ⊆ 𝑀 → ∅ ∈ 𝑀)
4		noel 4318	. . . . . 6 ⊢ ¬ 𝑧 ∈ ∅
5	4	rgenw 3054	. . . . 5 ⊢ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ ∅
6		eleq1 2821	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ∈ ω ↔ ∅ ∈ ω))
7		eleq2 2822	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∅))
8	7	notbid 318	. . . . . . . 8 ⊢ (𝑦 = ∅ → (¬ 𝑧 ∈ 𝑦 ↔ ¬ 𝑧 ∈ ∅))
9	8	ralbidv 3165	. . . . . . 7 ⊢ (𝑦 = ∅ → (∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦 ↔ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ ∅))
10	6, 9	anbi12d 632	. . . . . 6 ⊢ (𝑦 = ∅ → ((𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ↔ (∅ ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ ∅)))
11	10	rspcev 3605	. . . . 5 ⊢ ((∅ ∈ 𝑀 ∧ (∅ ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ ∅)) → ∃𝑦 ∈ 𝑀 (𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦))
12	1, 5, 11	mpanr12 705	. . . 4 ⊢ (∅ ∈ 𝑀 → ∃𝑦 ∈ 𝑀 (𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦))
13	3, 12	syl 17	. . 3 ⊢ (ω ⊆ 𝑀 → ∃𝑦 ∈ 𝑀 (𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦))
14		ssel 3957	. . . . . . 7 ⊢ (ω ⊆ 𝑀 → (suc 𝑦 ∈ ω → suc 𝑦 ∈ 𝑀))
15		peano2 7894	. . . . . . 7 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
16	14, 15	impel 505	. . . . . 6 ⊢ ((ω ⊆ 𝑀 ∧ 𝑦 ∈ ω) → suc 𝑦 ∈ 𝑀)
17	15	adantl 481	. . . . . 6 ⊢ ((ω ⊆ 𝑀 ∧ 𝑦 ∈ ω) → suc 𝑦 ∈ ω)
18		vex 3467	. . . . . . . . 9 ⊢ 𝑤 ∈ V
19	18	elsuc 6434	. . . . . . . 8 ⊢ (𝑤 ∈ suc 𝑦 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))
20	19	rgenw 3054	. . . . . . 7 ⊢ ∀𝑤 ∈ 𝑀 (𝑤 ∈ suc 𝑦 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))
21		eleq1 2821	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑦 → (𝑧 ∈ ω ↔ suc 𝑦 ∈ ω))
22		eleq2 2822	. . . . . . . . . . 11 ⊢ (𝑧 = suc 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ suc 𝑦))
23	22	bibi1d 343	. . . . . . . . . 10 ⊢ (𝑧 = suc 𝑦 → ((𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)) ↔ (𝑤 ∈ suc 𝑦 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))
24	23	ralbidv 3165	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑦 → (∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)) ↔ ∀𝑤 ∈ 𝑀 (𝑤 ∈ suc 𝑦 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))
25	21, 24	anbi12d 632	. . . . . . . 8 ⊢ (𝑧 = suc 𝑦 → ((𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))) ↔ (suc 𝑦 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ suc 𝑦 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
26	25	rspcev 3605	. . . . . . 7 ⊢ ((suc 𝑦 ∈ 𝑀 ∧ (suc 𝑦 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ suc 𝑦 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))) → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))
27	20, 26	mpanr2 704	. . . . . 6 ⊢ ((suc 𝑦 ∈ 𝑀 ∧ suc 𝑦 ∈ ω) → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))
28	16, 17, 27	syl2anc 584	. . . . 5 ⊢ ((ω ⊆ 𝑀 ∧ 𝑦 ∈ ω) → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))
29	28	ex 412	. . . 4 ⊢ (ω ⊆ 𝑀 → (𝑦 ∈ ω → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
30	29	ralrimivw 3137	. . 3 ⊢ (ω ⊆ 𝑀 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ ω → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
31		eleq2 2822	. . . . . . . 8 ⊢ (𝑥 = ω → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ω))
32	31	anbi1d 631	. . . . . . 7 ⊢ (𝑥 = ω → ((𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ↔ (𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦)))
33	32	rexbidv 3166	. . . . . 6 ⊢ (𝑥 = ω → (∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ↔ ∃𝑦 ∈ 𝑀 (𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦)))
34		eleq2 2822	. . . . . . . . . 10 ⊢ (𝑥 = ω → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ ω))
35	34	anbi1d 631	. . . . . . . . 9 ⊢ (𝑥 = ω → ((𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))) ↔ (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
36	35	rexbidv 3166	. . . . . . . 8 ⊢ (𝑥 = ω → (∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))) ↔ ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
37	31, 36	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = ω → ((𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))) ↔ (𝑦 ∈ ω → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))))
38	37	ralbidv 3165	. . . . . 6 ⊢ (𝑥 = ω → (∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))) ↔ ∀𝑦 ∈ 𝑀 (𝑦 ∈ ω → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))))
39	33, 38	anbi12d 632	. . . . 5 ⊢ (𝑥 = ω → ((∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))) ↔ (∃𝑦 ∈ 𝑀 (𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ ω → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))))
40	39	rspcev 3605	. . . 4 ⊢ ((ω ∈ 𝑀 ∧ (∃𝑦 ∈ 𝑀 (𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ ω → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))) → ∃𝑥 ∈ 𝑀 (∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))))
41	40	expcom 413	. . 3 ⊢ ((∃𝑦 ∈ 𝑀 (𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ ω → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))) → (ω ∈ 𝑀 → ∃𝑥 ∈ 𝑀 (∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))))
42	13, 30, 41	syl2anc 584	. 2 ⊢ (ω ⊆ 𝑀 → (ω ∈ 𝑀 → ∃𝑥 ∈ 𝑀 (∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))))
43	42	imp 406	1 ⊢ ((ω ⊆ 𝑀 ∧ ω ∈ 𝑀) → ∃𝑥 ∈ 𝑀 (∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ∧ ∀𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 → ∃𝑧 ∈ 𝑀 (𝑧 ∈ 𝑥 ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059   ⊆ wss 3931  ∅c0 4313  suc csuc 6365  ωcom 7869

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-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

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-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-tr 5240  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-om 7870

This theorem is referenced by:  omelaxinf2  44978