Theorem isinf 9213

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 5327. (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 5109	. . . . . 6 ⊢ (𝑛 = ∅ → (𝑥 ≈ 𝑛 ↔ 𝑥 ≈ ∅))
2	1	anbi2d 641	. . . . 5 ⊢ (𝑛 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
3	2	exbidv 1944	. . . 4 ⊢ (𝑛 = ∅ → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
4		breq2 5109	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑥 ≈ 𝑛 ↔ 𝑥 ≈ 𝑚))
5	4	anbi2d 641	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚)))
6	5	exbidv 1944	. . . 4 ⊢ (𝑛 = 𝑚 → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚)))
7		sseq1 3964	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
8	7	adantl 486	. . . . . 6 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦) → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
9		breq1 5108	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑛 ↔ 𝑦 ≈ 𝑛))
10		breq2 5109	. . . . . . 7 ⊢ (𝑛 = suc 𝑚 → (𝑦 ≈ 𝑛 ↔ 𝑦 ≈ suc 𝑚))
11	9, 10	sylan9bbr 519	. . . . . 6 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦) → (𝑥 ≈ 𝑛 ↔ 𝑦 ≈ suc 𝑚))
12	8, 11	anbi12d 643	. . . . 5 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
13	12	cbvexdvaw 2062	. . . 4 ⊢ (𝑛 = suc 𝑚 → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
14		0ss 4357	. . . . . 6 ⊢ ∅ ⊆ 𝐴
15		peano1 7873	. . . . . . 7 ⊢ ∅ ∈ ω
16		enrefnn 9031	. . . . . . 7 ⊢ (∅ ∈ ω → ∅ ≈ ∅)
17	15, 16	ax-mp 5	. . . . . 6 ⊢ ∅ ≈ ∅
18		0ex 5262	. . . . . . 7 ⊢ ∅ ∈ V
19		sseq1 3964	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
20		breq1 5108	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ↔ ∅ ≈ ∅))
21	19, 20	anbi12d 643	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ (∅ ⊆ 𝐴 ∧ ∅ ≈ ∅)))
22	18, 21	spcev 3568	. . . . . 6 ⊢ ((∅ ⊆ 𝐴 ∧ ∅ ≈ ∅) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅))
23	14, 17, 22	mp2an 704	. . . . 5 ⊢ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)
24	23	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅))
25		ssdif0 4322	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑥 ↔ (𝐴 ∖ 𝑥) = ∅)
26		eqss 3954	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 ↔ (𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥))
27		breq1 5108	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑚 ↔ 𝐴 ≈ 𝑚))
28	27	biimpa 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚) → 𝐴 ≈ 𝑚)
29		rspe 3255	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ 𝑚) → ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
30	28, 29	sylan2 604	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ω ∧ (𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚)) → ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
31		isfi 8960	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Fin ↔ ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
32	30, 31	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ω ∧ (𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚)) → 𝐴 ∈ Fin)
33	32	expcom 418	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → 𝐴 ∈ Fin))
34	26, 33	sylanbr 593	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥) ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → 𝐴 ∈ Fin))
35	34	ex 417	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥) → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → 𝐴 ∈ Fin)))
36	25, 35	sylan2br 606	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑥) = ∅) → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → 𝐴 ∈ Fin)))
37	36	expcom 418	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ 𝑥) = ∅ → (𝑥 ⊆ 𝐴 → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → 𝐴 ∈ Fin))))
38	37	3impd 1365	. . . . . . . . . 10 ⊢ ((𝐴 ∖ 𝑥) = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → 𝐴 ∈ Fin))
39	38	com12 33	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → ((𝐴 ∖ 𝑥) = ∅ → 𝐴 ∈ Fin))
40	39	con3d 153	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → (¬ 𝐴 ∈ Fin → ¬ (𝐴 ∖ 𝑥) = ∅))
41		bren 8941	. . . . . . . . . 10 ⊢ (𝑥 ≈ 𝑚 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑚)
42		neq0 4307	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐴 ∖ 𝑥) = ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 ∖ 𝑥))
43		eldifi 4087	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → 𝑧 ∈ 𝐴)
44	43	snssd 4748	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → {𝑧} ⊆ 𝐴)
45		unss 4145	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐴)
46	45	biimpi 219	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) → (𝑥 ∪ {𝑧}) ⊆ 𝐴)
47	44, 46	sylan2 604	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑥)) → (𝑥 ∪ {𝑧}) ⊆ 𝐴)
48	47	ad2ant2r 759	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑥 ∪ {𝑧}) ⊆ 𝐴)
49		vex 3461	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑧 ∈ V
50		vex 3461	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑚 ∈ V
51	49, 50	f1osn 6852	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ {⟨𝑧, 𝑚⟩}:{𝑧}–1-1-onto→{𝑚}
52	51	jctr 533	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑓:𝑥–1-1-onto→𝑚 → (𝑓:𝑥–1-1-onto→𝑚 ∧ {⟨𝑧, 𝑚⟩}:{𝑧}–1-1-onto→{𝑚}))
53		eldifn 4088	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → ¬ 𝑧 ∈ 𝑥)
54		disjsn 4673	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑥)
55	53, 54	sylibr 237	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → (𝑥 ∩ {𝑧}) = ∅)
56		nnord 7858	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ω → Ord 𝑚)
57		orddisj 6388	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (Ord 𝑚 → (𝑚 ∩ {𝑚}) = ∅)
58	56, 57	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑚 ∈ ω → (𝑚 ∩ {𝑚}) = ∅)
59	55, 58	anim12i 624	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω) → ((𝑥 ∩ {𝑧}) = ∅ ∧ (𝑚 ∩ {𝑚}) = ∅))
60		f1oun 6830	. . . . . . . . . . . . . . . . . . . . 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 607	. . . . . . . . . . . . . . . . . . . 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 6356	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
63		f1oeq3 6800	. . . . . . . . . . . . . . . . . . . . . 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 3461	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑓 ∈ V
66		snex 5401	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {⟨𝑧, 𝑚⟩} ∈ V
67	65, 66	unex 7731	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑓 ∪ {⟨𝑧, 𝑚⟩}) ∈ V
68		f1oeq1 6798	. . . . . . . . . . . . . . . . . . . . . . 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 3568	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 → ∃𝑔 𝑔:(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚)
70		bren 8941	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∪ {𝑧}) ≈ suc 𝑚 ↔ ∃𝑔 𝑔:(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚)
71	69, 70	sylibr 237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
72	64, 71	sylbir 238	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→(𝑚 ∪ {𝑚}) → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
73	61, 72	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:𝑥–1-1-onto→𝑚 ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
74	73	adantll 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
75		vex 3461	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑥 ∈ V
76		snex 5401	. . . . . . . . . . . . . . . . . . . 20 ⊢ {𝑧} ∈ V
77	75, 76	unex 7731	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∪ {𝑧}) ∈ V
78		sseq1 3964	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑦 ⊆ 𝐴 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐴))
79		breq1 5108	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑦 ≈ suc 𝑚 ↔ (𝑥 ∪ {𝑧}) ≈ suc 𝑚))
80	78, 79	anbi12d 643	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → ((𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚) ↔ ((𝑥 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∪ {𝑧}) ≈ suc 𝑚)))
81	77, 80	spcev 3568	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∪ {𝑧}) ≈ suc 𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))
82	48, 74, 81	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))
83	82	expcom 418	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω) → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
84	83	ex 417	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → (𝑚 ∈ ω → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
85	84	exlimiv 1953	. . . . . . . . . . . . . 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 220	. . . . . . . . . . . . 13 ⊢ (¬ (𝐴 ∖ 𝑥) = ∅ → (𝑚 ∈ ω → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
87	86	com13 89	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
88	87	expcom 418	. . . . . . . . . . 11 ⊢ (𝑓:𝑥–1-1-onto→𝑚 → (𝑥 ⊆ 𝐴 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
89	88	exlimiv 1953	. . . . . . . . . 10 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑚 → (𝑥 ⊆ 𝐴 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
90	41, 89	sylbi 220	. . . . . . . . 9 ⊢ (𝑥 ≈ 𝑚 → (𝑥 ⊆ 𝐴 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
91	90	3imp21 1129	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
92	40, 91	syld 48	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → (¬ 𝐴 ∈ Fin → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
93	92	3expia 1137	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → (¬ 𝐴 ∈ Fin → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
94	93	exlimiv 1953	. . . . 5 ⊢ (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → (¬ 𝐴 ∈ Fin → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
95	94	com3l 90	. . . 4 ⊢ (𝑚 ∈ ω → (¬ 𝐴 ∈ Fin → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
96	3, 6, 13, 24, 95	finds2 7883	. . 3 ⊢ (𝑛 ∈ ω → (¬ 𝐴 ∈ Fin → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)))
97	96	com12 33	. 2 ⊢ (¬ 𝐴 ∈ Fin → (𝑛 ∈ ω → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)))
98	97	ralrimiv 3156	1 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089   ∖ cdif 3904   ∪ cun 3905   ∩ cin 3906   ⊆ wss 3907  ∅c0 4288  {csn 4585  ⟨cop 4591   class class class wbr 5105  Ord word 6349  suc csuc 6352  –1-1-onto→wf1o 6524  ωcom 7850   ≈ cen 8928  Fincfn 8931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-om 7851  df-en 8932  df-fin 8935

This theorem is referenced by:  fineqvlem  9214  isinffi  9966  domtriomlem  10414  ishashinf  14490  prcinf  35421  ctbssinf  37912