Theorem omssaxinf2 45432

Description: A class that contains all ordinals up to and including ω models the Axiom of Infinity ax-inf2 9553. The antecedent of this theorem is not enough to guarantee that the class models the alternate axiom ax-inf 9550. (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 7829	. . . . 5 ⊢ ∅ ∈ ω
2		ssel 3909	. . . . 5 ⊢ (ω ⊆ 𝑀 → (∅ ∈ ω → ∅ ∈ 𝑀))
3	1, 2	mpi 20	. . . 4 ⊢ (ω ⊆ 𝑀 → ∅ ∈ 𝑀)
4		noel 4266	. . . . . 6 ⊢ ¬ 𝑧 ∈ ∅
5	4	rgenw 3057	. . . . 5 ⊢ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ ∅
6		eleq1 2827	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝑦 ∈ ω ↔ ∅ ∈ ω))
7		eleq2 2828	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∅))
8	7	notbid 319	. . . . . . . 8 ⊢ (𝑦 = ∅ → (¬ 𝑧 ∈ 𝑦 ↔ ¬ 𝑧 ∈ ∅))
9	8	ralbidv 3162	. . . . . . 7 ⊢ (𝑦 = ∅ → (∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦 ↔ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ ∅))
10	6, 9	anbi12d 638	. . . . . 6 ⊢ (𝑦 = ∅ → ((𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ↔ (∅ ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ ∅)))
11	10	rspcev 3560	. . . . 5 ⊢ ((∅ ∈ 𝑀 ∧ (∅ ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ ∅)) → ∃𝑦 ∈ 𝑀 (𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦))
12	1, 5, 11	mpanr12 711	. . . 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 3909	. . . . . . 7 ⊢ (ω ⊆ 𝑀 → (suc 𝑦 ∈ ω → suc 𝑦 ∈ 𝑀))
15		peano2 7830	. . . . . . 7 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
16	14, 15	impel 510	. . . . . 6 ⊢ ((ω ⊆ 𝑀 ∧ 𝑦 ∈ ω) → suc 𝑦 ∈ 𝑀)
17	15	adantl 482	. . . . . 6 ⊢ ((ω ⊆ 𝑀 ∧ 𝑦 ∈ ω) → suc 𝑦 ∈ ω)
18		vex 3435	. . . . . . . . 9 ⊢ 𝑤 ∈ V
19	18	elsuc 6382	. . . . . . . 8 ⊢ (𝑤 ∈ suc 𝑦 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))
20	19	rgenw 3057	. . . . . . 7 ⊢ ∀𝑤 ∈ 𝑀 (𝑤 ∈ suc 𝑦 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))
21		eleq1 2827	. . . . . . . . 9 ⊢ (𝑧 = suc 𝑦 → (𝑧 ∈ ω ↔ suc 𝑦 ∈ ω))
22		eleq2 2828	. . . . . . . . . . 11 ⊢ (𝑧 = suc 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ suc 𝑦))
23	22	bibi1d 344	. . . . . . . . . 10 ⊢ (𝑧 = suc 𝑦 → ((𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)) ↔ (𝑤 ∈ suc 𝑦 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))
24	23	ralbidv 3162	. . . . . . . . 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 638	. . . . . . . 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 3560	. . . . . . 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 710	. . . . . 6 ⊢ ((suc 𝑦 ∈ 𝑀 ∧ suc 𝑦 ∈ ω) → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))
28	16, 17, 27	syl2anc 590	. . . . 5 ⊢ ((ω ⊆ 𝑀 ∧ 𝑦 ∈ ω) → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦))))
29	28	ex 413	. . . 4 ⊢ (ω ⊆ 𝑀 → (𝑦 ∈ ω → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
30	29	ralrimivw 3135	. . 3 ⊢ (ω ⊆ 𝑀 → ∀𝑦 ∈ 𝑀 (𝑦 ∈ ω → ∃𝑧 ∈ 𝑀 (𝑧 ∈ ω ∧ ∀𝑤 ∈ 𝑀 (𝑤 ∈ 𝑧 ↔ (𝑤 ∈ 𝑦 ∨ 𝑤 = 𝑦)))))
31		eleq2 2828	. . . . . . . 8 ⊢ (𝑥 = ω → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ ω))
32	31	anbi1d 637	. . . . . . 7 ⊢ (𝑥 = ω → ((𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ↔ (𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦)))
33	32	rexbidv 3163	. . . . . 6 ⊢ (𝑥 = ω → (∃𝑦 ∈ 𝑀 (𝑦 ∈ 𝑥 ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦) ↔ ∃𝑦 ∈ 𝑀 (𝑦 ∈ ω ∧ ∀𝑧 ∈ 𝑀 ¬ 𝑧 ∈ 𝑦)))
34		eleq2 2828	. . . . . . . . . 10 ⊢ (𝑥 = ω → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ ω))
35	34	anbi1d 637	. . . . . . . . 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 3163	. . . . . . . 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 345	. . . . . . 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 3162	. . . . . 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 638	. . . . 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 3560	. . . 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 414	. . 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 590	. 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 407	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 207   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063   ⊆ wss 3883  ∅c0 4261  suc csuc 6312  ωcom 7806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-tr 5180  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-om 7807

This theorem is referenced by:  omelaxinf2  45433