Theorem isinf 8733

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.)

Assertion

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

Distinct variable group:   𝑥,𝐴,𝑛

Proof of Theorem isinf

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

Step	Hyp	Ref	Expression
1		breq2 5072	. . . . . 6 ⊢ (𝑛 = ∅ → (𝑥 ≈ 𝑛 ↔ 𝑥 ≈ ∅))
2	1	anbi2d 630	. . . . 5 ⊢ (𝑛 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
3	2	exbidv 1922	. . . 4 ⊢ (𝑛 = ∅ → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
4		breq2 5072	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑥 ≈ 𝑛 ↔ 𝑥 ≈ 𝑚))
5	4	anbi2d 630	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚)))
6	5	exbidv 1922	. . . 4 ⊢ (𝑛 = 𝑚 → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚)))
7		sseq1 3994	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
8	7	adantl 484	. . . . . 6 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦) → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
9		breq1 5071	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑛 ↔ 𝑦 ≈ 𝑛))
10		breq2 5072	. . . . . . 7 ⊢ (𝑛 = suc 𝑚 → (𝑦 ≈ 𝑛 ↔ 𝑦 ≈ suc 𝑚))
11	9, 10	sylan9bbr 513	. . . . . 6 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦) → (𝑥 ≈ 𝑛 ↔ 𝑦 ≈ suc 𝑚))
12	8, 11	anbi12d 632	. . . . 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 4352	. . . . . 6 ⊢ ∅ ⊆ 𝐴
15		0ex 5213	. . . . . . 7 ⊢ ∅ ∈ V
16	15	enref 8544	. . . . . 6 ⊢ ∅ ≈ ∅
17		sseq1 3994	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
18		breq1 5071	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ↔ ∅ ≈ ∅))
19	17, 18	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ (∅ ⊆ 𝐴 ∧ ∅ ≈ ∅)))
20	15, 19	spcev 3609	. . . . . 6 ⊢ ((∅ ⊆ 𝐴 ∧ ∅ ≈ ∅) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅))
21	14, 16, 20	mp2an 690	. . . . 5 ⊢ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)
22	21	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅))
23		ssdif0 4325	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑥 ↔ (𝐴 ∖ 𝑥) = ∅)
24		eqss 3984	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 ↔ (𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥))
25		breq1 5071	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑚 ↔ 𝐴 ≈ 𝑚))
26	25	biimpa 479	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚) → 𝐴 ≈ 𝑚)
27		rspe 3306	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ 𝑚) → ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
28	26, 27	sylan2 594	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ω ∧ (𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚)) → ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
29		isfi 8535	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Fin ↔ ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
30	28, 29	sylibr 236	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ω ∧ (𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚)) → 𝐴 ∈ Fin)
31	30	expcom 416	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → 𝐴 ∈ Fin))
32	24, 31	sylanbr 584	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥) ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → 𝐴 ∈ Fin))
33	32	ex 415	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥) → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → 𝐴 ∈ Fin)))
34	23, 33	sylan2br 596	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑥) = ∅) → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → 𝐴 ∈ Fin)))
35	34	expcom 416	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ 𝑥) = ∅ → (𝑥 ⊆ 𝐴 → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → 𝐴 ∈ Fin))))
36	35	3impd 1344	. . . . . . . . . 10 ⊢ ((𝐴 ∖ 𝑥) = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → 𝐴 ∈ Fin))
37	36	com12 32	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → ((𝐴 ∖ 𝑥) = ∅ → 𝐴 ∈ Fin))
38	37	con3d 155	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → (¬ 𝐴 ∈ Fin → ¬ (𝐴 ∖ 𝑥) = ∅))
39		bren 8520	. . . . . . . . . 10 ⊢ (𝑥 ≈ 𝑚 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑚)
40		neq0 4311	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐴 ∖ 𝑥) = ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 ∖ 𝑥))
41		eldifi 4105	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → 𝑧 ∈ 𝐴)
42	41	snssd 4744	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → {𝑧} ⊆ 𝐴)
43		unss 4162	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐴)
44	43	biimpi 218	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) → (𝑥 ∪ {𝑧}) ⊆ 𝐴)
45	42, 44	sylan2 594	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑥)) → (𝑥 ∪ {𝑧}) ⊆ 𝐴)
46	45	ad2ant2r 745	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑥 ∪ {𝑧}) ⊆ 𝐴)
47		vex 3499	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑧 ∈ V
48		vex 3499	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑚 ∈ V
49	47, 48	f1osn 6656	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {⟨𝑧, 𝑚⟩}:{𝑧}–1-1-onto→{𝑚}
50	49	jctr 527	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:𝑥–1-1-onto→𝑚 → (𝑓:𝑥–1-1-onto→𝑚 ∧ {⟨𝑧, 𝑚⟩}:{𝑧}–1-1-onto→{𝑚}))
51		eldifn 4106	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → ¬ 𝑧 ∈ 𝑥)
52		disjsn 4649	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑥)
53	51, 52	sylibr 236	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → (𝑥 ∩ {𝑧}) = ∅)
54		nnord 7590	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ω → Ord 𝑚)
55		orddisj 6231	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Ord 𝑚 → (𝑚 ∩ {𝑚}) = ∅)
56	54, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ω → (𝑚 ∩ {𝑚}) = ∅)
57	53, 56	anim12i 614	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω) → ((𝑥 ∩ {𝑧}) = ∅ ∧ (𝑚 ∩ {𝑚}) = ∅))
58		f1oun 6636	. . . . . . . . . . . . . . . . . . . . 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}))
59	50, 57, 58	syl2an 597	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:𝑥–1-1-onto→𝑚 ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→(𝑚 ∪ {𝑚}))
60		df-suc 6199	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
61		f1oeq3 6608	. . . . . . . . . . . . . . . . . . . . . 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})))
62	60, 61	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 ↔ (𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→(𝑚 ∪ {𝑚}))
63		vex 3499	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑓 ∈ V
64		snex 5334	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {⟨𝑧, 𝑚⟩} ∈ V
65	63, 64	unex 7471	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∪ {⟨𝑧, 𝑚⟩}) ∈ V
66		f1oeq1 6606	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = (𝑓 ∪ {⟨𝑧, 𝑚⟩}) → (𝑔:(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 ↔ (𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚))
67	65, 66	spcev 3609	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 → ∃𝑔 𝑔:(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚)
68		bren 8520	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∪ {𝑧}) ≈ suc 𝑚 ↔ ∃𝑔 𝑔:(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚)
69	67, 68	sylibr 236	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
70	62, 69	sylbir 237	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→(𝑚 ∪ {𝑚}) → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
71	59, 70	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝑥–1-1-onto→𝑚 ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
72	71	adantll 712	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
73		vex 3499	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
74		snex 5334	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑧} ∈ V
75	73, 74	unex 7471	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∪ {𝑧}) ∈ V
76		sseq1 3994	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑦 ⊆ 𝐴 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐴))
77		breq1 5071	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑦 ≈ suc 𝑚 ↔ (𝑥 ∪ {𝑧}) ≈ suc 𝑚))
78	76, 77	anbi12d 632	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → ((𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚) ↔ ((𝑥 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∪ {𝑧}) ≈ suc 𝑚)))
79	75, 78	spcev 3609	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∪ {𝑧}) ≈ suc 𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))
80	46, 72, 79	syl2anc 586	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))
81	80	expcom 416	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω) → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
82	81	ex 415	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → (𝑚 ∈ ω → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
83	82	exlimiv 1931	. . . . . . . . . . . . . 14 ⊢ (∃𝑧 𝑧 ∈ (𝐴 ∖ 𝑥) → (𝑚 ∈ ω → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
84	40, 83	sylbi 219	. . . . . . . . . . . . 13 ⊢ (¬ (𝐴 ∖ 𝑥) = ∅ → (𝑚 ∈ ω → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
85	84	com13 88	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
86	85	expcom 416	. . . . . . . . . . 11 ⊢ (𝑓:𝑥–1-1-onto→𝑚 → (𝑥 ⊆ 𝐴 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
87	86	exlimiv 1931	. . . . . . . . . 10 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑚 → (𝑥 ⊆ 𝐴 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
88	39, 87	sylbi 219	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝑚 → (𝑥 ⊆ 𝐴 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
89	88	3imp21 1110	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
90	38, 89	syld 47	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → (¬ 𝐴 ∈ Fin → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
91	90	3expia 1117	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → (¬ 𝐴 ∈ Fin → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
92	91	exlimiv 1931	. . . . 5 ⊢ (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → (¬ 𝐴 ∈ Fin → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
93	92	com3l 89	. . . 4 ⊢ (𝑚 ∈ ω → (¬ 𝐴 ∈ Fin → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
94	3, 6, 13, 22, 93	finds2 7612	. . 3 ⊢ (𝑛 ∈ ω → (¬ 𝐴 ∈ Fin → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)))
95	94	com12 32	. 2 ⊢ (¬ 𝐴 ∈ Fin → (𝑛 ∈ ω → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)))
96	95	ralrimiv 3183	1 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ∖ cdif 3935   ∪ cun 3936   ∩ cin 3937   ⊆ wss 3938  ∅c0 4293  {csn 4569  ⟨cop 4575   class class class wbr 5068  Ord word 6192  suc csuc 6195  –1-1-onto→wf1o 6356  ωcom 7582   ≈ cen 8508  Fincfn 8511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-om 7583  df-en 8512  df-fin 8515

This theorem is referenced by:  fineqvlem  8734  isinffi  9423  domtriomlem  9866  ishashinf  13824  ctbssinf  34689