Theorem ctbssinf 35504

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 8965	. 2 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
2		omex 9331	. . 3 ⊢ ω ∈ V
3		sseq1 3942	. . . 4 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ⊆ 𝐴 ↔ (𝑓‘𝑛) ⊆ 𝐴))
4		breq1 5073	. . . 4 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ≈ 𝑛 ↔ (𝑓‘𝑛) ≈ 𝑛))
5	3, 4	anbi12d 630	. . 3 ⊢ (𝑥 = (𝑓‘𝑛) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)))
6	2, 5	ac6s2 10173	. 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 3086	. . . . 5 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴)
9		fvex 6769	. . . . . . . 8 ⊢ (𝑓‘𝑛) ∈ V
10	9	elpw 4534	. . . . . . 7 ⊢ ((𝑓‘𝑛) ∈ 𝒫 𝐴 ↔ (𝑓‘𝑛) ⊆ 𝐴)
11	10	ralbii 3090	. . . . . 6 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛) ∈ 𝒫 𝐴 ↔ ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴)
12		fnfvrnss 6976	. . . . . . 7 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓‘𝑛) ∈ 𝒫 𝐴) → ran 𝑓 ⊆ 𝒫 𝐴)
13		uniss 4844	. . . . . . . 8 ⊢ (ran 𝑓 ⊆ 𝒫 𝐴 → ∪ ran 𝑓 ⊆ ∪ 𝒫 𝐴)
14		unipw 5360	. . . . . . . 8 ⊢ ∪ 𝒫 𝐴 = 𝐴
15	13, 14	sseqtrdi 3967	. . . . . . 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 6810	. . . . . . . . . . 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 5836	. . . . . . . . 9 ⊢ (𝑓 Fn ω → ran 𝑓 = ran (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
22	21	unieqd 4850	. . . . . . . 8 ⊢ (𝑓 Fn ω → ∪ ran 𝑓 = ∪ ran (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
23	9	dfiun3 5864	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ω (𝑓‘𝑛) = ∪ ran (𝑛 ∈ ω ↦ (𝑓‘𝑛))
24	22, 23	eqtr4di 2797	. . . . . . 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 3086	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓‘𝑛) ≈ 𝑛)
28		endom 8722	. . . . . . . . . 10 ⊢ ((𝑓‘𝑛) ≈ 𝑛 → (𝑓‘𝑛) ≼ 𝑛)
29		nnsdom 9342	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → 𝑛 ≺ ω)
30		domsdomtr 8848	. . . . . . . . . . 11 ⊢ (((𝑓‘𝑛) ≼ 𝑛 ∧ 𝑛 ≺ ω) → (𝑓‘𝑛) ≺ ω)
31		sdomdom 8723	. . . . . . . . . . 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 3085	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛) ≈ 𝑛 → ∀𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
35		iunctb2 35501	. . . . . . . 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 5092	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ≼ ω)
39		fvssunirn 6785	. . . . . . . . . 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 3086	. . . . . . 7 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
43		sseq1 3942	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ⊆ ∪ ran 𝑓 ↔ (𝑓‘𝑛) ⊆ ∪ ran 𝑓))
44	43, 4	anbi12d 630	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑓‘𝑛) → ((𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛) ↔ ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛)))
45	9, 44	spcev 3535	. . . . . . . . . 10 ⊢ (((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∃𝑥(𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛))
46	45	ralimi 3086	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛))
47		isinf2 35503	. . . . . . . . 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 3426	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
50	49	rnex 7733	. . . . . . . . . 10 ⊢ ran 𝑓 ∈ V
51	50	uniex 7572	. . . . . . . . 9 ⊢ ∪ ran 𝑓 ∈ V
52		infinf 10253	. . . . . . . . 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 481	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ω ≼ ∪ ran 𝑓)
57		sbth 8833	. . . . 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 3942	. . . . . 6 ⊢ (𝑥 = ∪ ran 𝑓 → (𝑥 ⊆ 𝐴 ↔ ∪ ran 𝑓 ⊆ 𝐴))
60		breq1 5073	. . . . . 6 ⊢ (𝑥 = ∪ ran 𝑓 → (𝑥 ≈ ω ↔ ∪ ran 𝑓 ≈ ω))
61	59, 60	anbi12d 630	. . . . 5 ⊢ (𝑥 = ∪ ran 𝑓 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω) ↔ (∪ ran 𝑓 ⊆ 𝐴 ∧ ∪ ran 𝑓 ≈ ω)))
62	51, 61	spcev 3535	. . . 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 1934	. 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 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836  ∪ ciun 4921   class class class wbr 5070   ↦ cmpt 5153  ran crn 5581   Fn wfn 6413  ‘cfv 6418  ωcom 7687   ≈ cen 8688   ≼ cdom 8689   ≺ csdm 8690  Fincfn 8691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-reg 9281  ax-inf2 9329  ax-cc 10122  ax-ac2 10150

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-oi 9199  df-r1 9453  df-rank 9454  df-card 9628  df-acn 9631  df-ac 9803

This theorem is referenced by: (None)