Theorem ctbssinf 34690

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 8731	. 2 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
2		omex 9106	. . 3 ⊢ ω ∈ V
3		sseq1 3992	. . . 4 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ⊆ 𝐴 ↔ (𝑓‘𝑛) ⊆ 𝐴))
4		breq1 5069	. . . 4 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ≈ 𝑛 ↔ (𝑓‘𝑛) ≈ 𝑛))
5	3, 4	anbi12d 632	. . 3 ⊢ (𝑥 = (𝑓‘𝑛) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)))
6	2, 5	ac6s2 9908	. 2 ⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ∃𝑓(𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)))
7		simpl 485	. . . . . 6 ⊢ (((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → (𝑓‘𝑛) ⊆ 𝐴)
8	7	ralimi 3160	. . . . 5 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴)
9		fvex 6683	. . . . . . . 8 ⊢ (𝑓‘𝑛) ∈ V
10	9	elpw 4543	. . . . . . 7 ⊢ ((𝑓‘𝑛) ∈ 𝒫 𝐴 ↔ (𝑓‘𝑛) ⊆ 𝐴)
11	10	ralbii 3165	. . . . . 6 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛) ∈ 𝒫 𝐴 ↔ ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴)
12		fnfvrnss 6884	. . . . . . 7 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓‘𝑛) ∈ 𝒫 𝐴) → ran 𝑓 ⊆ 𝒫 𝐴)
13		uniss 4846	. . . . . . . 8 ⊢ (ran 𝑓 ⊆ 𝒫 𝐴 → ∪ ran 𝑓 ⊆ ∪ 𝒫 𝐴)
14		unipw 5343	. . . . . . . 8 ⊢ ∪ 𝒫 𝐴 = 𝐴
15	13, 14	sseqtrdi 4017	. . . . . . 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 596	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴) → ∪ ran 𝑓 ⊆ 𝐴)
18	8, 17	sylan2 594	. . . 4 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ⊆ 𝐴)
19		dffn5 6724	. . . . . . . . . . 11 ⊢ (𝑓 Fn ω ↔ 𝑓 = (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
20	19	biimpi 218	. . . . . . . . . 10 ⊢ (𝑓 Fn ω → 𝑓 = (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
21	20	rneqd 5808	. . . . . . . . 9 ⊢ (𝑓 Fn ω → ran 𝑓 = ran (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
22	21	unieqd 4852	. . . . . . . 8 ⊢ (𝑓 Fn ω → ∪ ran 𝑓 = ∪ ran (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
23	9	dfiun3 5837	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ω (𝑓‘𝑛) = ∪ ran (𝑛 ∈ ω ↦ (𝑓‘𝑛))
24	22, 23	syl6eqr 2874	. . . . . . 7 ⊢ (𝑓 Fn ω → ∪ ran 𝑓 = ∪ 𝑛 ∈ ω (𝑓‘𝑛))
25	24	adantr 483	. . . . . 6 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 = ∪ 𝑛 ∈ ω (𝑓‘𝑛))
26		simpr 487	. . . . . . . . 9 ⊢ (((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → (𝑓‘𝑛) ≈ 𝑛)
27	26	ralimi 3160	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓‘𝑛) ≈ 𝑛)
28		endom 8536	. . . . . . . . . 10 ⊢ ((𝑓‘𝑛) ≈ 𝑛 → (𝑓‘𝑛) ≼ 𝑛)
29		nnsdom 9117	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → 𝑛 ≺ ω)
30		domsdomtr 8652	. . . . . . . . . . 11 ⊢ (((𝑓‘𝑛) ≼ 𝑛 ∧ 𝑛 ≺ ω) → (𝑓‘𝑛) ≺ ω)
31		sdomdom 8537	. . . . . . . . . . 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 598	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ∧ (𝑓‘𝑛) ≈ 𝑛) → (𝑓‘𝑛) ≼ ω)
34	33	ralimiaa 3159	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛) ≈ 𝑛 → ∀𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
35		iunctb2 34687	. . . . . . . 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 484	. . . . . 6 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ 𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
38	25, 37	eqbrtrd 5088	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ≼ ω)
39		fvssunirn 6699	. . . . . . . . . 10 ⊢ (𝑓‘𝑛) ⊆ ∪ ran 𝑓
40	39	jctl 526	. . . . . . . . 9 ⊢ ((𝑓‘𝑛) ≈ 𝑛 → ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
41	40	adantl 484	. . . . . . . 8 ⊢ (((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
42	41	ralimi 3160	. . . . . . 7 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
43		sseq1 3992	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ⊆ ∪ ran 𝑓 ↔ (𝑓‘𝑛) ⊆ ∪ ran 𝑓))
44	43, 4	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑓‘𝑛) → ((𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛) ↔ ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛)))
45	9, 44	spcev 3607	. . . . . . . . . 10 ⊢ (((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∃𝑥(𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛))
46	45	ralimi 3160	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛))
47		isinf2 34689	. . . . . . . . 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 3497	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
50	49	rnex 7617	. . . . . . . . . 10 ⊢ ran 𝑓 ∈ V
51	50	uniex 7467	. . . . . . . . 9 ⊢ ∪ ran 𝑓 ∈ V
52		infinf 9988	. . . . . . . . 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 220	. . . . . . 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 484	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ω ≼ ∪ ran 𝑓)
57		sbth 8637	. . . . 5 ⊢ ((∪ ran 𝑓 ≼ ω ∧ ω ≼ ∪ ran 𝑓) → ∪ ran 𝑓 ≈ ω)
58	38, 56, 57	syl2anc 586	. . . 4 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ≈ ω)
59		sseq1 3992	. . . . . 6 ⊢ (𝑥 = ∪ ran 𝑓 → (𝑥 ⊆ 𝐴 ↔ ∪ ran 𝑓 ⊆ 𝐴))
60		breq1 5069	. . . . . 6 ⊢ (𝑥 = ∪ ran 𝑓 → (𝑥 ≈ ω ↔ ∪ ran 𝑓 ≈ ω))
61	59, 60	anbi12d 632	. . . . 5 ⊢ (𝑥 = ∪ ran 𝑓 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω) ↔ (∪ ran 𝑓 ⊆ 𝐴 ∧ ∪ ran 𝑓 ≈ ω)))
62	51, 61	spcev 3607	. . . 4 ⊢ ((∪ ran 𝑓 ⊆ 𝐴 ∧ ∪ ran 𝑓 ≈ ω) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
63	18, 58, 62	syl2anc 586	. . 3 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
64	63	exlimiv 1931	. 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 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3138  Vcvv 3494   ⊆ wss 3936  𝒫 cpw 4539  ∪ cuni 4838  ∪ ciun 4919   class class class wbr 5066   ↦ cmpt 5146  ran crn 5556   Fn wfn 6350  ‘cfv 6355  ωcom 7580   ≈ cen 8506   ≼ cdom 8507   ≺ csdm 8508  Fincfn 8509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-reg 9056  ax-inf2 9104  ax-cc 9857  ax-ac2 9885

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-oi 8974  df-r1 9193  df-rank 9194  df-card 9368  df-acn 9371  df-ac 9542

This theorem is referenced by: (None)