Theorem ctbssinf 37372

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 9323	. 2 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
2		omex 9712	. . 3 ⊢ ω ∈ V
3		sseq1 4034	. . . 4 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ⊆ 𝐴 ↔ (𝑓‘𝑛) ⊆ 𝐴))
4		breq1 5169	. . . 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 10555	. 2 ⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ∃𝑓(𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)))
7		simpl 482	. . . . . 6 ⊢ (((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → (𝑓‘𝑛) ⊆ 𝐴)
8	7	ralimi 3089	. . . . 5 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴)
9		fvex 6933	. . . . . . . 8 ⊢ (𝑓‘𝑛) ∈ V
10	9	elpw 4626	. . . . . . 7 ⊢ ((𝑓‘𝑛) ∈ 𝒫 𝐴 ↔ (𝑓‘𝑛) ⊆ 𝐴)
11	10	ralbii 3099	. . . . . 6 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛) ∈ 𝒫 𝐴 ↔ ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴)
12		fnfvrnss 7155	. . . . . . 7 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓‘𝑛) ∈ 𝒫 𝐴) → ran 𝑓 ⊆ 𝒫 𝐴)
13		uniss 4939	. . . . . . . 8 ⊢ (ran 𝑓 ⊆ 𝒫 𝐴 → ∪ ran 𝑓 ⊆ ∪ 𝒫 𝐴)
14		unipw 5470	. . . . . . . 8 ⊢ ∪ 𝒫 𝐴 = 𝐴
15	13, 14	sseqtrdi 4059	. . . . . . 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 594	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴) → ∪ ran 𝑓 ⊆ 𝐴)
18	8, 17	sylan2 592	. . . 4 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ⊆ 𝐴)
19		dffn5 6980	. . . . . . . . . . 11 ⊢ (𝑓 Fn ω ↔ 𝑓 = (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
20	19	biimpi 216	. . . . . . . . . 10 ⊢ (𝑓 Fn ω → 𝑓 = (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
21	20	rneqd 5963	. . . . . . . . 9 ⊢ (𝑓 Fn ω → ran 𝑓 = ran (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
22	21	unieqd 4944	. . . . . . . 8 ⊢ (𝑓 Fn ω → ∪ ran 𝑓 = ∪ ran (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
23	9	dfiun3 5992	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ω (𝑓‘𝑛) = ∪ ran (𝑛 ∈ ω ↦ (𝑓‘𝑛))
24	22, 23	eqtr4di 2798	. . . . . . 7 ⊢ (𝑓 Fn ω → ∪ ran 𝑓 = ∪ 𝑛 ∈ ω (𝑓‘𝑛))
25	24	adantr 480	. . . . . 6 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 = ∪ 𝑛 ∈ ω (𝑓‘𝑛))
26		simpr 484	. . . . . . . . 9 ⊢ (((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → (𝑓‘𝑛) ≈ 𝑛)
27	26	ralimi 3089	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓‘𝑛) ≈ 𝑛)
28		endom 9039	. . . . . . . . . 10 ⊢ ((𝑓‘𝑛) ≈ 𝑛 → (𝑓‘𝑛) ≼ 𝑛)
29		nnsdom 9723	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → 𝑛 ≺ ω)
30		domsdomtr 9178	. . . . . . . . . . 11 ⊢ (((𝑓‘𝑛) ≼ 𝑛 ∧ 𝑛 ≺ ω) → (𝑓‘𝑛) ≺ ω)
31		sdomdom 9040	. . . . . . . . . . 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 596	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ∧ (𝑓‘𝑛) ≈ 𝑛) → (𝑓‘𝑛) ≼ ω)
34	33	ralimiaa 3088	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛) ≈ 𝑛 → ∀𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
35		iunctb2 37369	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ 𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
38	25, 37	eqbrtrd 5188	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ≼ ω)
39		fvssunirn 6953	. . . . . . . . . 10 ⊢ (𝑓‘𝑛) ⊆ ∪ ran 𝑓
40	39	jctl 523	. . . . . . . . 9 ⊢ ((𝑓‘𝑛) ≈ 𝑛 → ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
41	40	adantl 481	. . . . . . . 8 ⊢ (((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
42	41	ralimi 3089	. . . . . . 7 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
43		sseq1 4034	. . . . . . . . . . . 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 3619	. . . . . . . . . 10 ⊢ (((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∃𝑥(𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛))
46	45	ralimi 3089	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛))
47		isinf2 37371	. . . . . . . . 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 3492	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
50	49	rnex 7950	. . . . . . . . . 10 ⊢ ran 𝑓 ∈ V
51	50	uniex 7776	. . . . . . . . 9 ⊢ ∪ ran 𝑓 ∈ V
52		infinf 10635	. . . . . . . . 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 218	. . . . . . 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 481	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ω ≼ ∪ ran 𝑓)
57		sbth 9159	. . . . 5 ⊢ ((∪ ran 𝑓 ≼ ω ∧ ω ≼ ∪ ran 𝑓) → ∪ ran 𝑓 ≈ ω)
58	38, 56, 57	syl2anc 583	. . . 4 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ≈ ω)
59		sseq1 4034	. . . . . 6 ⊢ (𝑥 = ∪ ran 𝑓 → (𝑥 ⊆ 𝐴 ↔ ∪ ran 𝑓 ⊆ 𝐴))
60		breq1 5169	. . . . . 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 3619	. . . 4 ⊢ ((∪ ran 𝑓 ⊆ 𝐴 ∧ ∪ ran 𝑓 ≈ ω) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
63	18, 58, 62	syl2anc 583	. . 3 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
64	63	exlimiv 1929	. 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 206   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931  ∪ ciun 5015   class class class wbr 5166   ↦ cmpt 5249  ran crn 5701   Fn wfn 6568  ‘cfv 6573  ωcom 7903   ≈ cen 9000   ≼ cdom 9001   ≺ csdm 9002  Fincfn 9003

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-reg 9661  ax-inf2 9710  ax-cc 10504  ax-ac2 10532

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-oi 9579  df-r1 9833  df-rank 9834  df-card 10008  df-acn 10011  df-ac 10185

This theorem is referenced by: (None)