Theorem ctbssinf 35950

Description: Using the axiom of choice, any infinite class has a countable subset. (Contributed by ML, 14-Dec-2020.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem ctbssinf

Dummy variables 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isinf 9211	. 2 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
2		omex 9588	. . 3 ⊢ ω ∈ V
3		sseq1 3972	. . . 4 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ⊆ 𝐴 ↔ (𝑓‘𝑛) ⊆ 𝐴))
4		breq1 5113	. . . 4 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ≈ 𝑛 ↔ (𝑓‘𝑛) ≈ 𝑛))
5	3, 4	anbi12d 631	. . 3 ⊢ (𝑥 = (𝑓‘𝑛) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)))
6	2, 5	ac6s2 10431	. 2 ⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ∃𝑓(𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)))
7		simpl 483	. . . . . 6 ⊢ (((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → (𝑓‘𝑛) ⊆ 𝐴)
8	7	ralimi 3082	. . . . 5 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴)
9		fvex 6860	. . . . . . . 8 ⊢ (𝑓‘𝑛) ∈ V
10	9	elpw 4569	. . . . . . 7 ⊢ ((𝑓‘𝑛) ∈ 𝒫 𝐴 ↔ (𝑓‘𝑛) ⊆ 𝐴)
11	10	ralbii 3092	. . . . . 6 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛) ∈ 𝒫 𝐴 ↔ ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴)
12		fnfvrnss 7073	. . . . . . 7 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓‘𝑛) ∈ 𝒫 𝐴) → ran 𝑓 ⊆ 𝒫 𝐴)
13		uniss 4878	. . . . . . . 8 ⊢ (ran 𝑓 ⊆ 𝒫 𝐴 → ∪ ran 𝑓 ⊆ ∪ 𝒫 𝐴)
14		unipw 5412	. . . . . . . 8 ⊢ ∪ 𝒫 𝐴 = 𝐴
15	13, 14	sseqtrdi 3997	. . . . . . 7 ⊢ (ran 𝑓 ⊆ 𝒫 𝐴 → ∪ ran 𝑓 ⊆ 𝐴)
16	12, 15	syl 17	. . . . . 6 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓‘𝑛) ∈ 𝒫 𝐴) → ∪ ran 𝑓 ⊆ 𝐴)
17	11, 16	sylan2br 595	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴) → ∪ ran 𝑓 ⊆ 𝐴)
18	8, 17	sylan2 593	. . . 4 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ⊆ 𝐴)
19		dffn5 6906	. . . . . . . . . . 11 ⊢ (𝑓 Fn ω ↔ 𝑓 = (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
20	19	biimpi 215	. . . . . . . . . 10 ⊢ (𝑓 Fn ω → 𝑓 = (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
21	20	rneqd 5898	. . . . . . . . 9 ⊢ (𝑓 Fn ω → ran 𝑓 = ran (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
22	21	unieqd 4884	. . . . . . . 8 ⊢ (𝑓 Fn ω → ∪ ran 𝑓 = ∪ ran (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
23	9	dfiun3 5926	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ω (𝑓‘𝑛) = ∪ ran (𝑛 ∈ ω ↦ (𝑓‘𝑛))
24	22, 23	eqtr4di 2789	. . . . . . 7 ⊢ (𝑓 Fn ω → ∪ ran 𝑓 = ∪ 𝑛 ∈ ω (𝑓‘𝑛))
25	24	adantr 481	. . . . . 6 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 = ∪ 𝑛 ∈ ω (𝑓‘𝑛))
26		simpr 485	. . . . . . . . 9 ⊢ (((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → (𝑓‘𝑛) ≈ 𝑛)
27	26	ralimi 3082	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓‘𝑛) ≈ 𝑛)
28		endom 8926	. . . . . . . . . 10 ⊢ ((𝑓‘𝑛) ≈ 𝑛 → (𝑓‘𝑛) ≼ 𝑛)
29		nnsdom 9599	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → 𝑛 ≺ ω)
30		domsdomtr 9063	. . . . . . . . . . 11 ⊢ (((𝑓‘𝑛) ≼ 𝑛 ∧ 𝑛 ≺ ω) → (𝑓‘𝑛) ≺ ω)
31		sdomdom 8927	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑛) ≺ ω → (𝑓‘𝑛) ≼ ω)
32	30, 31	syl 17	. . . . . . . . . 10 ⊢ (((𝑓‘𝑛) ≼ 𝑛 ∧ 𝑛 ≺ ω) → (𝑓‘𝑛) ≼ ω)
33	28, 29, 32	syl2anr 597	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ∧ (𝑓‘𝑛) ≈ 𝑛) → (𝑓‘𝑛) ≼ ω)
34	33	ralimiaa 3081	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛) ≈ 𝑛 → ∀𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
35		iunctb2 35947	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛) ≼ ω → ∪ 𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
36	27, 34, 35	3syl 18	. . . . . . 7 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∪ 𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
37	36	adantl 482	. . . . . 6 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ 𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
38	25, 37	eqbrtrd 5132	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ≼ ω)
39		fvssunirn 6880	. . . . . . . . . 10 ⊢ (𝑓‘𝑛) ⊆ ∪ ran 𝑓
40	39	jctl 524	. . . . . . . . 9 ⊢ ((𝑓‘𝑛) ≈ 𝑛 → ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
41	40	adantl 482	. . . . . . . 8 ⊢ (((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
42	41	ralimi 3082	. . . . . . 7 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
43		sseq1 3972	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ⊆ ∪ ran 𝑓 ↔ (𝑓‘𝑛) ⊆ ∪ ran 𝑓))
44	43, 4	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑓‘𝑛) → ((𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛) ↔ ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛)))
45	9, 44	spcev 3566	. . . . . . . . . 10 ⊢ (((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∃𝑥(𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛))
46	45	ralimi 3082	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛))
47		isinf2 35949	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛) → ¬ ∪ ran 𝑓 ∈ Fin)
48	46, 47	syl 17	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ¬ ∪ ran 𝑓 ∈ Fin)
49		vex 3450	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
50	49	rnex 7854	. . . . . . . . . 10 ⊢ ran 𝑓 ∈ V
51	50	uniex 7683	. . . . . . . . 9 ⊢ ∪ ran 𝑓 ∈ V
52		infinf 10511	. . . . . . . . 9 ⊢ (∪ ran 𝑓 ∈ V → (¬ ∪ ran 𝑓 ∈ Fin ↔ ω ≼ ∪ ran 𝑓))
53	51, 52	ax-mp 5	. . . . . . . 8 ⊢ (¬ ∪ ran 𝑓 ∈ Fin ↔ ω ≼ ∪ ran 𝑓)
54	48, 53	sylib 217	. . . . . . 7 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ω ≼ ∪ ran 𝑓)
55	42, 54	syl 17	. . . . . 6 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ω ≼ ∪ ran 𝑓)
56	55	adantl 482	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ω ≼ ∪ ran 𝑓)
57		sbth 9044	. . . . 5 ⊢ ((∪ ran 𝑓 ≼ ω ∧ ω ≼ ∪ ran 𝑓) → ∪ ran 𝑓 ≈ ω)
58	38, 56, 57	syl2anc 584	. . . 4 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ≈ ω)
59		sseq1 3972	. . . . . 6 ⊢ (𝑥 = ∪ ran 𝑓 → (𝑥 ⊆ 𝐴 ↔ ∪ ran 𝑓 ⊆ 𝐴))
60		breq1 5113	. . . . . 6 ⊢ (𝑥 = ∪ ran 𝑓 → (𝑥 ≈ ω ↔ ∪ ran 𝑓 ≈ ω))
61	59, 60	anbi12d 631	. . . . 5 ⊢ (𝑥 = ∪ ran 𝑓 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω) ↔ (∪ ran 𝑓 ⊆ 𝐴 ∧ ∪ ran 𝑓 ≈ ω)))
62	51, 61	spcev 3566	. . . 4 ⊢ ((∪ ran 𝑓 ⊆ 𝐴 ∧ ∪ ran 𝑓 ≈ ω) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
63	18, 58, 62	syl2anc 584	. . 3 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
64	63	exlimiv 1933	. 2 ⊢ (∃𝑓(𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
65	1, 6, 64	3syl 18	1 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∀wral 3060  Vcvv 3446   ⊆ wss 3913  𝒫 cpw 4565  ∪ cuni 4870  ∪ ciun 4959   class class class wbr 5110   ↦ cmpt 5193  ran crn 5639   Fn wfn 6496  ‘cfv 6501  ωcom 7807   ≈ cen 8887   ≼ cdom 8888   ≺ csdm 8889  Fincfn 8890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-reg 9537  ax-inf2 9586  ax-cc 10380  ax-ac2 10408

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-oi 9455  df-r1 9709  df-rank 9710  df-card 9884  df-acn 9887  df-ac 10061

This theorem is referenced by: (None)