Theorem isinf 8329

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 4790	. . . . . 6 ⊢ (𝑛 = ∅ → (𝑥 ≈ 𝑛 ↔ 𝑥 ≈ ∅))
2	1	anbi2d 614	. . . . 5 ⊢ (𝑛 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
3	2	exbidv 2002	. . . 4 ⊢ (𝑛 = ∅ → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)))
4		breq2 4790	. . . . . 6 ⊢ (𝑛 = 𝑚 → (𝑥 ≈ 𝑛 ↔ 𝑥 ≈ 𝑚))
5	4	anbi2d 614	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ (𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚)))
6	5	exbidv 2002	. . . 4 ⊢ (𝑛 = 𝑚 → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚)))
7		sseq1 3775	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
8	7	adantl 467	. . . . . 6 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦) → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴))
9		breq1 4789	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ≈ 𝑛 ↔ 𝑦 ≈ 𝑛))
10		breq2 4790	. . . . . . 7 ⊢ (𝑛 = suc 𝑚 → (𝑦 ≈ 𝑛 ↔ 𝑦 ≈ suc 𝑚))
11	9, 10	sylan9bbr 500	. . . . . 6 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦) → (𝑥 ≈ 𝑛 ↔ 𝑦 ≈ suc 𝑚))
12	8, 11	anbi12d 616	. . . . 5 ⊢ ((𝑛 = suc 𝑚 ∧ 𝑥 = 𝑦) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ (𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
13	12	cbvexdva 2441	. . . 4 ⊢ (𝑛 = suc 𝑚 → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
14		0ss 4116	. . . . . 6 ⊢ ∅ ⊆ 𝐴
15		0ex 4924	. . . . . . 7 ⊢ ∅ ∈ V
16	15	enref 8142	. . . . . 6 ⊢ ∅ ≈ ∅
17		sseq1 3775	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
18		breq1 4789	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥 ≈ ∅ ↔ ∅ ≈ ∅))
19	17, 18	anbi12d 616	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅) ↔ (∅ ⊆ 𝐴 ∧ ∅ ≈ ∅)))
20	15, 19	spcev 3451	. . . . . 6 ⊢ ((∅ ⊆ 𝐴 ∧ ∅ ≈ ∅) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅))
21	14, 16, 20	mp2an 672	. . . . 5 ⊢ ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅)
22	21	a1i 11	. . . 4 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ∅))
23		ssdif0 4089	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ 𝑥 ↔ (𝐴 ∖ 𝑥) = ∅)
24		eqss 3767	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝐴 ↔ (𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥))
25		breq1 4789	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝐴 → (𝑥 ≈ 𝑚 ↔ 𝐴 ≈ 𝑚))
26	25	biimpa 462	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚) → 𝐴 ≈ 𝑚)
27		rspe 3151	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ 𝑚) → ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
28	26, 27	sylan2 580	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ ω ∧ (𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚)) → ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
29		isfi 8133	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Fin ↔ ∃𝑚 ∈ ω 𝐴 ≈ 𝑚)
30	28, 29	sylibr 224	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ ω ∧ (𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚)) → 𝐴 ∈ Fin)
31	30	expcom 398	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → 𝐴 ∈ Fin))
32	24, 31	sylanbr 571	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥) ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → 𝐴 ∈ Fin))
33	32	ex 397	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝐴 ⊆ 𝑥) → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → 𝐴 ∈ Fin)))
34	23, 33	sylan2br 582	. . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ 𝐴 ∧ (𝐴 ∖ 𝑥) = ∅) → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → 𝐴 ∈ Fin)))
35	34	expcom 398	. . . . . . . . . . 11 ⊢ ((𝐴 ∖ 𝑥) = ∅ → (𝑥 ⊆ 𝐴 → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → 𝐴 ∈ Fin))))
36	35	3impd 1441	. . . . . . . . . 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 149	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → (¬ 𝐴 ∈ Fin → ¬ (𝐴 ∖ 𝑥) = ∅))
39		bren 8118	. . . . . . . . . . 11 ⊢ (𝑥 ≈ 𝑚 ↔ ∃𝑓 𝑓:𝑥–1-1-onto→𝑚)
40		neq0 4077	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝐴 ∖ 𝑥) = ∅ ↔ ∃𝑧 𝑧 ∈ (𝐴 ∖ 𝑥))
41		eldifi 3883	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → 𝑧 ∈ 𝐴)
42	41	snssd 4475	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → {𝑧} ⊆ 𝐴)
43		unss 3938	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐴)
44	43	biimpi 206	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ⊆ 𝐴 ∧ {𝑧} ⊆ 𝐴) → (𝑥 ∪ {𝑧}) ⊆ 𝐴)
45	42, 44	sylan2 580	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑧 ∈ (𝐴 ∖ 𝑥)) → (𝑥 ∪ {𝑧}) ⊆ 𝐴)
46	45	ad2ant2r 741	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑥 ∪ {𝑧}) ⊆ 𝐴)
47		vex 3354	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑧 ∈ V
48		vex 3354	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑚 ∈ V
49	47, 48	f1osn 6317	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {⟨𝑧, 𝑚⟩}:{𝑧}–1-1-onto→{𝑚}
50	49	jctr 514	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑓:𝑥–1-1-onto→𝑚 → (𝑓:𝑥–1-1-onto→𝑚 ∧ {⟨𝑧, 𝑚⟩}:{𝑧}–1-1-onto→{𝑚}))
51		eldifn 3884	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → ¬ 𝑧 ∈ 𝑥)
52		disjsn 4383	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑥 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑥)
53	51, 52	sylibr 224	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → (𝑥 ∩ {𝑧}) = ∅)
54		nnord 7220	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ ω → Ord 𝑚)
55		orddisj 5905	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (Ord 𝑚 → (𝑚 ∩ {𝑚}) = ∅)
56	54, 55	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 ∈ ω → (𝑚 ∩ {𝑚}) = ∅)
57	53, 56	anim12i 600	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω) → ((𝑥 ∩ {𝑧}) = ∅ ∧ (𝑚 ∩ {𝑚}) = ∅))
58		f1oun 6297	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((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 583	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓:𝑥–1-1-onto→𝑚 ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→(𝑚 ∪ {𝑚}))
60		df-suc 5872	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ suc 𝑚 = (𝑚 ∪ {𝑚})
61		f1oeq3 6270	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (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	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 ↔ (𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→(𝑚 ∪ {𝑚}))
63		vex 3354	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝑓 ∈ V
64		snex 5036	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ {⟨𝑧, 𝑚⟩} ∈ V
65	63, 64	unex 7103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑓 ∪ {⟨𝑧, 𝑚⟩}) ∈ V
66		f1oeq1 6268	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (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 3451	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 → ∃𝑔 𝑔:(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚)
68		bren 8118	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑥 ∪ {𝑧}) ≈ suc 𝑚 ↔ ∃𝑔 𝑔:(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚)
69	67, 68	sylibr 224	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→suc 𝑚 → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
70	62, 69	sylbir 225	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 ∪ {⟨𝑧, 𝑚⟩}):(𝑥 ∪ {𝑧})–1-1-onto→(𝑚 ∪ {𝑚}) → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
71	59, 70	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:𝑥–1-1-onto→𝑚 ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
72	71	adantll 693	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → (𝑥 ∪ {𝑧}) ≈ suc 𝑚)
73		vex 3354	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑥 ∈ V
74		snex 5036	. . . . . . . . . . . . . . . . . . . . 21 ⊢ {𝑧} ∈ V
75	73, 74	unex 7103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∪ {𝑧}) ∈ V
76		sseq1 3775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑦 ⊆ 𝐴 ↔ (𝑥 ∪ {𝑧}) ⊆ 𝐴))
77		breq1 4789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → (𝑦 ≈ suc 𝑚 ↔ (𝑥 ∪ {𝑧}) ≈ suc 𝑚))
78	76, 77	anbi12d 616	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑥 ∪ {𝑧}) → ((𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚) ↔ ((𝑥 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∪ {𝑧}) ≈ suc 𝑚)))
79	75, 78	spcev 3451	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 ∪ {𝑧}) ⊆ 𝐴 ∧ (𝑥 ∪ {𝑧}) ≈ suc 𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))
80	46, 72, 79	syl2anc 573	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) ∧ (𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω)) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))
81	80	expcom 398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (𝐴 ∖ 𝑥) ∧ 𝑚 ∈ ω) → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
82	81	ex 397	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ (𝐴 ∖ 𝑥) → (𝑚 ∈ ω → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
83	82	exlimiv 2010	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧 𝑧 ∈ (𝐴 ∖ 𝑥) → (𝑚 ∈ ω → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
84	40, 83	sylbi 207	. . . . . . . . . . . . . 14 ⊢ (¬ (𝐴 ∖ 𝑥) = ∅ → (𝑚 ∈ ω → ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
85	84	com13 88	. . . . . . . . . . . . 13 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑓:𝑥–1-1-onto→𝑚) → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
86	85	expcom 398	. . . . . . . . . . . 12 ⊢ (𝑓:𝑥–1-1-onto→𝑚 → (𝑥 ⊆ 𝐴 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
87	86	exlimiv 2010	. . . . . . . . . . 11 ⊢ (∃𝑓 𝑓:𝑥–1-1-onto→𝑚 → (𝑥 ⊆ 𝐴 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
88	39, 87	sylbi 207	. . . . . . . . . 10 ⊢ (𝑥 ≈ 𝑚 → (𝑥 ⊆ 𝐴 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
89	88	com12 32	. . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐴 → (𝑥 ≈ 𝑚 → (𝑚 ∈ ω → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))))
90	89	3imp 1101	. . . . . . . 8 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → (¬ (𝐴 ∖ 𝑥) = ∅ → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
91	38, 90	syld 47	. . . . . . 7 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚 ∧ 𝑚 ∈ ω) → (¬ 𝐴 ∈ Fin → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚)))
92	91	3expia 1114	. . . . . 6 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → (¬ 𝐴 ∈ Fin → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
93	92	exlimiv 2010	. . . . 5 ⊢ (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚) → (𝑚 ∈ ω → (¬ 𝐴 ∈ Fin → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
94	93	com3l 89	. . . 4 ⊢ (𝑚 ∈ ω → (¬ 𝐴 ∈ Fin → (∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑚) → ∃𝑦(𝑦 ⊆ 𝐴 ∧ 𝑦 ≈ suc 𝑚))))
95	3, 6, 13, 22, 94	finds2 7241	. . 3 ⊢ (𝑛 ∈ ω → (¬ 𝐴 ∈ Fin → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)))
96	95	com12 32	. 2 ⊢ (¬ 𝐴 ∈ Fin → (𝑛 ∈ ω → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛)))
97	96	ralrimiv 3114	1 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631  ∃wex 1852   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062   ∖ cdif 3720   ∪ cun 3721   ∩ cin 3722   ⊆ wss 3723  ∅c0 4063  {csn 4316  ⟨cop 4322   class class class wbr 4786  Ord word 5865  suc csuc 5868  –1-1-onto→wf1o 6030  ωcom 7212   ≈ cen 8106  Fincfn 8109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-om 7213  df-en 8110  df-fin 8113

This theorem is referenced by:  fineqvlem  8330  isinffi  9018  domtriomlem  9466  ishashinf  13449