Theorem ctbssinf 38249

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 9234	. 2 ⊢ (¬ 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛))
2		omex 9622	. . 3 ⊢ ω ∈ V
3		sseq1 3955	. . . 4 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ⊆ 𝐴 ↔ (𝑓‘𝑛) ⊆ 𝐴))
4		breq1 5105	. . . 4 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ≈ 𝑛 ↔ (𝑓‘𝑛) ≈ 𝑛))
5	3, 4	anbi12d 644	. . 3 ⊢ (𝑥 = (𝑓‘𝑛) → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) ↔ ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)))
6	2, 5	ac6s2 10536	. 2 ⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ 𝑛) → ∃𝑓(𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)))
7		simpl 488	. . . . . 6 ⊢ (((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → (𝑓‘𝑛) ⊆ 𝐴)
8	7	ralimi 3099	. . . . 5 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴)
9		fvex 6886	. . . . . . . 8 ⊢ (𝑓‘𝑛) ∈ V
10	9	elpw 4560	. . . . . . 7 ⊢ ((𝑓‘𝑛) ∈ 𝒫 𝐴 ↔ (𝑓‘𝑛) ⊆ 𝐴)
11	10	ralbii 3108	. . . . . 6 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛) ∈ 𝒫 𝐴 ↔ ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴)
12		fnfvrnss 7109	. . . . . . 7 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓‘𝑛) ∈ 𝒫 𝐴) → ran 𝑓 ⊆ 𝒫 𝐴)
13		uniss 4874	. . . . . . . 8 ⊢ (ran 𝑓 ⊆ 𝒫 𝐴 → ∪ ran 𝑓 ⊆ ∪ 𝒫 𝐴)
14		unipw 5417	. . . . . . . 8 ⊢ ∪ 𝒫 𝐴 = 𝐴
15	13, 14	sseqtrdi 3970	. . . . . . 7 ⊢ (ran 𝑓 ⊆ 𝒫 𝐴 → ∪ ran 𝑓 ⊆ 𝐴)
16	12, 15	syl 18	. . . . . 6 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓‘𝑛) ∈ 𝒫 𝐴) → ∪ ran 𝑓 ⊆ 𝐴)
17	11, 16	sylan2br 607	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓‘𝑛) ⊆ 𝐴) → ∪ ran 𝑓 ⊆ 𝐴)
18	8, 17	sylan2 605	. . . 4 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ⊆ 𝐴)
19		dffn5 6931	. . . . . . . . . . 11 ⊢ (𝑓 Fn ω ↔ 𝑓 = (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
20	19	biimpi 219	. . . . . . . . . 10 ⊢ (𝑓 Fn ω → 𝑓 = (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
21	20	rneqd 5916	. . . . . . . . 9 ⊢ (𝑓 Fn ω → ran 𝑓 = ran (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
22	21	unieqd 4879	. . . . . . . 8 ⊢ (𝑓 Fn ω → ∪ ran 𝑓 = ∪ ran (𝑛 ∈ ω ↦ (𝑓‘𝑛)))
23	9	dfiun3 5948	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ ω (𝑓‘𝑛) = ∪ ran (𝑛 ∈ ω ↦ (𝑓‘𝑛))
24	22, 23	eqtr4di 2813	. . . . . . 7 ⊢ (𝑓 Fn ω → ∪ ran 𝑓 = ∪ 𝑛 ∈ ω (𝑓‘𝑛))
25	24	adantr 486	. . . . . 6 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 = ∪ 𝑛 ∈ ω (𝑓‘𝑛))
26		simpr 490	. . . . . . . . 9 ⊢ (((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → (𝑓‘𝑛) ≈ 𝑛)
27	26	ralimi 3099	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓‘𝑛) ≈ 𝑛)
28		endom 8984	. . . . . . . . . 10 ⊢ ((𝑓‘𝑛) ≈ 𝑛 → (𝑓‘𝑛) ≼ 𝑛)
29		nnsdom 9633	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → 𝑛 ≺ ω)
30		domsdomtr 9109	. . . . . . . . . . 11 ⊢ (((𝑓‘𝑛) ≼ 𝑛 ∧ 𝑛 ≺ ω) → (𝑓‘𝑛) ≺ ω)
31		sdomdom 8985	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑛) ≺ ω → (𝑓‘𝑛) ≼ ω)
32	30, 31	syl 18	. . . . . . . . . 10 ⊢ (((𝑓‘𝑛) ≼ 𝑛 ∧ 𝑛 ≺ ω) → (𝑓‘𝑛) ≼ ω)
33	28, 29, 32	syl2anr 609	. . . . . . . . 9 ⊢ ((𝑛 ∈ ω ∧ (𝑓‘𝑛) ≈ 𝑛) → (𝑓‘𝑛) ≼ ω)
34	33	ralimiaa 3098	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛) ≈ 𝑛 → ∀𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
35		iunctb2 38246	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛) ≼ ω → ∪ 𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
36	27, 34, 35	3syl 19	. . . . . . 7 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∪ 𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
37	36	adantl 487	. . . . . 6 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ 𝑛 ∈ ω (𝑓‘𝑛) ≼ ω)
38	25, 37	eqbrtrd 5126	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ≼ ω)
39		fvssunirn 6904	. . . . . . . . . 10 ⊢ (𝑓‘𝑛) ⊆ ∪ ran 𝑓
40	39	jctl 533	. . . . . . . . 9 ⊢ ((𝑓‘𝑛) ≈ 𝑛 → ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
41	40	adantl 487	. . . . . . . 8 ⊢ (((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
42	41	ralimi 3099	. . . . . . 7 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛))
43		sseq1 3955	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘𝑛) → (𝑥 ⊆ ∪ ran 𝑓 ↔ (𝑓‘𝑛) ⊆ ∪ ran 𝑓))
44	43, 4	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑓‘𝑛) → ((𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛) ↔ ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛)))
45	9, 44	spcev 3560	. . . . . . . . . 10 ⊢ (((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∃𝑥(𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛))
46	45	ralimi 3099	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛))
47		isinf2 38248	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω ∃𝑥(𝑥 ⊆ ∪ ran 𝑓 ∧ 𝑥 ≈ 𝑛) → ¬ ∪ ran 𝑓 ∈ Fin)
48	46, 47	syl 18	. . . . . . . 8 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ¬ ∪ ran 𝑓 ∈ Fin)
49		vex 3454	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
50	49	rnex 7905	. . . . . . . . . 10 ⊢ ran 𝑓 ∈ V
51	50	uniex 7741	. . . . . . . . 9 ⊢ ∪ ran 𝑓 ∈ V
52		infinf 10623	. . . . . . . . 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 221	. . . . . . 7 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ ∪ ran 𝑓 ∧ (𝑓‘𝑛) ≈ 𝑛) → ω ≼ ∪ ran 𝑓)
55	42, 54	syl 18	. . . . . 6 ⊢ (∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛) → ω ≼ ∪ ran 𝑓)
56	55	adantl 487	. . . . 5 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ω ≼ ∪ ran 𝑓)
57		sbth 9094	. . . . 5 ⊢ ((∪ ran 𝑓 ≼ ω ∧ ω ≼ ∪ ran 𝑓) → ∪ ran 𝑓 ≈ ω)
58	38, 56, 57	syl2anc 596	. . . 4 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∪ ran 𝑓 ≈ ω)
59		sseq1 3955	. . . . . 6 ⊢ (𝑥 = ∪ ran 𝑓 → (𝑥 ⊆ 𝐴 ↔ ∪ ran 𝑓 ⊆ 𝐴))
60		breq1 5105	. . . . . 6 ⊢ (𝑥 = ∪ ran 𝑓 → (𝑥 ≈ ω ↔ ∪ ran 𝑓 ≈ ω))
61	59, 60	anbi12d 644	. . . . 5 ⊢ (𝑥 = ∪ ran 𝑓 → ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω) ↔ (∪ ran 𝑓 ⊆ 𝐴 ∧ ∪ ran 𝑓 ≈ ω)))
62	51, 61	spcev 3560	. . . 4 ⊢ ((∪ ran 𝑓 ⊆ 𝐴 ∧ ∪ ran 𝑓 ≈ ω) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
63	18, 58, 62	syl2anc 596	. . 3 ⊢ ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
64	63	exlimiv 1963	. 2 ⊢ (∃𝑓(𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓‘𝑛) ⊆ 𝐴 ∧ (𝑓‘𝑛) ≈ 𝑛)) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))
65	1, 6, 64	3syl 19	1 ⊢ (¬ 𝐴 ∈ Fin → ∃𝑥(𝑥 ⊆ 𝐴 ∧ 𝑥 ≈ ω))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ⊆ wss 3898  𝒫 cpw 4556  ∪ cuni 4866  ∪ ciun 4950   class class class wbr 5102   ↦ cmpt 5185  ran crn 5648   Fn wfn 6522  ‘cfv 6527  ωcom 7860   ≈ cen 8948   ≼ cdom 8949   ≺ csdm 8950  Fincfn 8951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-reg 9564  ax-inf2 9620  ax-cc 10485  ax-ac2 10513

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-er 8695  df-map 8827  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-oi 9482  df-r1 9746  df-rank 9747  df-scott 9901  df-card 9992  df-acn 9995  df-ac 10167

This theorem is used by: (None)