Theorem isinf 9172

Description: Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by Mario Carneiro, 15-Jan-2013.) Avoid ax-pow 5301. (Revised by BTernaryTau, 2-Jan-2025.)

Assertion

Ref	Expression
isinf	⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))

Distinct variable group:   𝐴,𝑛,𝑥

Proof of Theorem isinf

Dummy variables 𝑚 𝑦 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5083	. . . . . 6 ⊢ (𝑛 = ∅ → (𝑥 ≈ 𝑛 ↔ 𝑥 ≈ ∅))
2	1	anbi2d 636	. . . . 5 ⊢ (𝑛 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
3	2	exbidv 1928	. . . 4 ⊢ (𝑛 = ∅ → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
4		breq2 5083	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑥 ≈ 𝑛 ↔ 𝑥 ≈ 𝑚))
5	4	anbi2d 636	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚)))
6	5	exbidv 1928	. . . 4 ⊢ (𝑛 = 𝑚 → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚)))
7		sseq1 3947	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
8	7	adantl 482	. . . . . 6 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦) → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
9		breq1 5082	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑛 ↔ 𝑦 ≈ 𝑛))
10		breq2 5083	. . . . . . 7 ⊢ (𝑛 = suc 𝑚 → (𝑦 ≈ 𝑛 ↔ 𝑦 ≈ suc 𝑚))
11	9, 10	sylan9bbr 515	. . . . . 6 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦) → (𝑥 ≈ 𝑛 ↔ 𝑦 ≈ suc 𝑚))
12	8, 11	anbi12d 638	. . . . 5 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
13	12	cbvexdvaw 2046	. . . 4 ⊢ (𝑛 = suc 𝑚 → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
14		0ss 4335	. . . . . 6 ⊢ ∅ ⊆ 𝐴
15		peano1 7836	. . . . . . 7 ⊢ ∅ ∈ ω
16		enrefnn 8990	. . . . . . 7 ⊢ (∅ ∈ ω → ∅ ≈ ∅)
17	15, 16	ax-mp 5	. . . . . 6 ⊢ ∅ ≈ ∅
18		0ex 5236	. . . . . . 7 ⊢ ∅ ∈ V
19		sseq1 3947	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
20		breq1 5082	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ↔ ∅ ≈ ∅))
21	19, 20	anbi12d 638	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ (∅ ⊆ 𝐴 ∧ ∅ ≈ ∅)))
22	18, 21	spcev 3551	. . . . . 6 ⊢ ((∅ ⊆ 𝐴 ∧ ∅ ≈ ∅) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅))
23	14, 17, 22	mp2an 698	. . . . 5 ⊢ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)
24	23	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅))
25		ssdif0 4301	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑥 ↔ (𝐴 ∖ 𝑥) = ∅)
26		eqss 3937	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 ↔ (𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥))
27		breq1 5082	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑚 ↔ 𝐴 ≈ 𝑚))
28	27	biimpa 477	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚) → 𝐴 ≈ 𝑚)
29		rspe 3230	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ 𝑚) → ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
30	28, 29	sylan2 599	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ω ∧ (𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚)) → ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
31		isfi 8919	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Fin ↔ ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
32	30, 31	sylibr 235	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ω ∧ (𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚)) → 𝐴 ∈ Fin)
33	32	expcom 414	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → 𝐴 ∈ Fin))
34	26, 33	sylanbr 588	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥) ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → 𝐴 ∈ Fin))
35	34	ex 413	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥) → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → 𝐴 ∈ Fin)))
36	25, 35	sylan2br 601	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑥) = ∅) → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → 𝐴 ∈ Fin)))
37	36	expcom 414	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ 𝑥) = ∅ → (𝑥 ⊆ 𝐴 → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → 𝐴 ∈ Fin))))
38	37	3impd 1355	. . . . . . . . . 10 ⊢ ((𝐴 ∖ 𝑥) = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → 𝐴 ∈ Fin))
39	38	com12 32	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → ((𝐴 ∖ 𝑥) = ∅ → 𝐴 ∈ Fin))
40	39	con3d 152	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → (¬ 𝐴 ∈ Fin → ¬ (𝐴 ∖ 𝑥) = ∅))
41		bren 8900	. . . . . . . . . 10 ⊢ (𝑥 ≈ 𝑚 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑚)
42		neq0 4287	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐴 ∖ 𝑥) = ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 ∖ 𝑥))
43		eldifi 4068	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → 𝑧 ∈ 𝐴)
44	43	snssd 4725	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → {𝑧} ⊆ 𝐴)
45		unss 4126	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐴)
46	45	biimpi 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) → (𝑥 ∪ {𝑧}) ⊆ 𝐴)
47	44, 46	sylan2 599	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑥)) → (𝑥 ∪ {𝑧}) ⊆ 𝐴)
48	47	ad2ant2r 753	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑥 ∪ {𝑧}) ⊆ 𝐴)
49		vex 3436	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑧 ∈ V
50		vex 3436	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑚 ∈ V
51	49, 50	f1osn 6815	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {⟨𝑧, 𝑚⟩}:{𝑧}–1-1-onto→{𝑚}
52	51	jctr 529	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:𝑥–1-1-onto→𝑚 → (𝑓:𝑥–1-1-onto→𝑚 ∧ {⟨𝑧, 𝑚⟩}:{𝑧}–1-1-onto→{𝑚}))
53		eldifn 4069	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → ¬ 𝑧 ∈ 𝑥)
54		disjsn 4650	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑥)
55	53, 54	sylibr 235	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → (𝑥 ∩ {𝑧}) = ∅)
56		nnord 7821	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ω → Ord 𝑚)
57		orddisj 6355	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Ord 𝑚 → (𝑚 ∩ {𝑚}) = ∅)
58	56, 57	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ω → (𝑚 ∩ {𝑚}) = ∅)
59	55, 58	anim12i 619	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω) → ((𝑥 ∩ {𝑧}) = ∅ ∧ (𝑚 ∩ {𝑚}) = ∅))
60		f1oun 6793	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓:𝑥–1-1-onto→𝑚 ∧ {⟨𝑧, 𝑚⟩}:{𝑧}–1-1-onto→{𝑚}) ∧ ((𝑥 ∩ {𝑧}) = ∅ ∧ (𝑚 ∩ {𝑚}) = ∅)) → (𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→(𝑚 ∪ {𝑚}))
61	52, 59, 60	syl2an 602	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:𝑥–1-1-onto→𝑚 ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→(𝑚 ∪ {𝑚}))
62		df-suc 6323	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
63		f1oeq3 6764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (suc 𝑚 = (𝑚 ∪ {𝑚}) → ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 ↔ (𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→(𝑚 ∪ {𝑚})))
64	62, 63	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 ↔ (𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→(𝑚 ∪ {𝑚}))
65		vex 3436	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑓 ∈ V
66		snex 5375	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {⟨𝑧, 𝑚⟩} ∈ V
67	65, 66	unex 7694	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∪ {⟨𝑧, 𝑚⟩}) ∈ V
68		f1oeq1 6762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑚⟩}) → (𝑔:(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 ↔ (𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚))
69	67, 68	spcev 3551	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 → ∃𝑔 𝑔:(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚)
70		bren 8900	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∪ {𝑧}) ≈ suc 𝑚 ↔ ∃𝑔 𝑔:(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚)
71	69, 70	sylibr 235	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
72	64, 71	sylbir 236	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→(𝑚 ∪ {𝑚}) → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
73	61, 72	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝑥–1-1-onto→𝑚 ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
74	73	adantll 720	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
75		vex 3436	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
76		snex 5375	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑧} ∈ V
77	75, 76	unex 7694	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∪ {𝑧}) ∈ V
78		sseq1 3947	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑦 ⊆ 𝐴 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐴))
79		breq1 5082	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑦 ≈ suc 𝑚 ↔ (𝑥 ∪ {𝑧}) ≈ suc 𝑚))
80	78, 79	anbi12d 638	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → ((𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚) ↔ ((𝑥 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∪ {𝑧}) ≈ suc 𝑚)))
81	77, 80	spcev 3551	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∪ {𝑧}) ≈ suc 𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))
82	48, 74, 81	syl2anc 590	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))
83	82	expcom 414	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω) → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
84	83	ex 413	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → (𝑚 ∈ ω → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
85	84	exlimiv 1937	. . . . . . . . . . . . . 14 ⊢ (∃𝑧 𝑧 ∈ (𝐴 ∖ 𝑥) → (𝑚 ∈ ω → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
86	42, 85	sylbi 218	. . . . . . . . . . . . 13 ⊢ (¬ (𝐴 ∖ 𝑥) = ∅ → (𝑚 ∈ ω → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
87	86	com13 88	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
88	87	expcom 414	. . . . . . . . . . 11 ⊢ (𝑓:𝑥–1-1-onto→𝑚 → (𝑥 ⊆ 𝐴 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
89	88	exlimiv 1937	. . . . . . . . . 10 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑚 → (𝑥 ⊆ 𝐴 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
90	41, 89	sylbi 218	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝑚 → (𝑥 ⊆ 𝐴 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
91	90	3imp21 1119	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
92	40, 91	syld 47	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → (¬ 𝐴 ∈ Fin → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
93	92	3expia 1127	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → (¬ 𝐴 ∈ Fin → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
94	93	exlimiv 1937	. . . . 5 ⊢ (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → (¬ 𝐴 ∈ Fin → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
95	94	com3l 89	. . . 4 ⊢ (𝑚 ∈ ω → (¬ 𝐴 ∈ Fin → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
96	3, 6, 13, 24, 95	finds2 7845	. . 3 ⊢ (𝑛 ∈ ω → (¬ 𝐴 ∈ Fin → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)))
97	96	com12 32	. 2 ⊢ (¬ 𝐴 ∈ Fin → (𝑛 ∈ ω → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)))
98	97	ralrimiv 3131	1 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547  ∃wex 1786   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890  ∅c0 4268  {csn 4562  ⟨cop 4568   class class class wbr 5079  Ord word 6316  suc csuc 6319  –1-1-onto→wf1o 6491  ωcom 7813   ≈ cen 8887  Fincfn 8890

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-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

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-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-om 7814  df-en 8891  df-fin 8894

This theorem is referenced by:  fineqvlem  9173  isinffi  9914  domtriomlem  10362  ishashinf  14423  prcinf  35304  ctbssinf  37769