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Theorem ctbssinf 34824
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(𝑥𝐴𝑥 ≈ ω))
Distinct variable group:   𝑥,𝐴

Proof of Theorem ctbssinf
Dummy variables 𝑓 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isinf 8719 . 2 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛))
2 omex 9094 . . 3 ω ∈ V
3 sseq1 3943 . . . 4 (𝑥 = (𝑓𝑛) → (𝑥𝐴 ↔ (𝑓𝑛) ⊆ 𝐴))
4 breq1 5036 . . . 4 (𝑥 = (𝑓𝑛) → (𝑥𝑛 ↔ (𝑓𝑛) ≈ 𝑛))
53, 4anbi12d 633 . . 3 (𝑥 = (𝑓𝑛) → ((𝑥𝐴𝑥𝑛) ↔ ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)))
62, 5ac6s2 9901 . 2 (∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛) → ∃𝑓(𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)))
7 simpl 486 . . . . . 6 (((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → (𝑓𝑛) ⊆ 𝐴)
87ralimi 3131 . . . . 5 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓𝑛) ⊆ 𝐴)
9 fvex 6662 . . . . . . . 8 (𝑓𝑛) ∈ V
109elpw 4504 . . . . . . 7 ((𝑓𝑛) ∈ 𝒫 𝐴 ↔ (𝑓𝑛) ⊆ 𝐴)
1110ralbii 3136 . . . . . 6 (∀𝑛 ∈ ω (𝑓𝑛) ∈ 𝒫 𝐴 ↔ ∀𝑛 ∈ ω (𝑓𝑛) ⊆ 𝐴)
12 fnfvrnss 6865 . . . . . . 7 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓𝑛) ∈ 𝒫 𝐴) → ran 𝑓 ⊆ 𝒫 𝐴)
13 uniss 4811 . . . . . . . 8 (ran 𝑓 ⊆ 𝒫 𝐴 ran 𝑓 𝒫 𝐴)
14 unipw 5311 . . . . . . . 8 𝒫 𝐴 = 𝐴
1513, 14sseqtrdi 3968 . . . . . . 7 (ran 𝑓 ⊆ 𝒫 𝐴 ran 𝑓𝐴)
1612, 15syl 17 . . . . . 6 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓𝑛) ∈ 𝒫 𝐴) → ran 𝑓𝐴)
1711, 16sylan2br 597 . . . . 5 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓𝑛) ⊆ 𝐴) → ran 𝑓𝐴)
188, 17sylan2 595 . . . 4 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ran 𝑓𝐴)
19 dffn5 6703 . . . . . . . . . . 11 (𝑓 Fn ω ↔ 𝑓 = (𝑛 ∈ ω ↦ (𝑓𝑛)))
2019biimpi 219 . . . . . . . . . 10 (𝑓 Fn ω → 𝑓 = (𝑛 ∈ ω ↦ (𝑓𝑛)))
2120rneqd 5776 . . . . . . . . 9 (𝑓 Fn ω → ran 𝑓 = ran (𝑛 ∈ ω ↦ (𝑓𝑛)))
2221unieqd 4817 . . . . . . . 8 (𝑓 Fn ω → ran 𝑓 = ran (𝑛 ∈ ω ↦ (𝑓𝑛)))
239dfiun3 5806 . . . . . . . 8 𝑛 ∈ ω (𝑓𝑛) = ran (𝑛 ∈ ω ↦ (𝑓𝑛))
2422, 23eqtr4di 2854 . . . . . . 7 (𝑓 Fn ω → ran 𝑓 = 𝑛 ∈ ω (𝑓𝑛))
2524adantr 484 . . . . . 6 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ran 𝑓 = 𝑛 ∈ ω (𝑓𝑛))
26 simpr 488 . . . . . . . . 9 (((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → (𝑓𝑛) ≈ 𝑛)
2726ralimi 3131 . . . . . . . 8 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓𝑛) ≈ 𝑛)
28 endom 8523 . . . . . . . . . 10 ((𝑓𝑛) ≈ 𝑛 → (𝑓𝑛) ≼ 𝑛)
29 nnsdom 9105 . . . . . . . . . 10 (𝑛 ∈ ω → 𝑛 ≺ ω)
30 domsdomtr 8640 . . . . . . . . . . 11 (((𝑓𝑛) ≼ 𝑛𝑛 ≺ ω) → (𝑓𝑛) ≺ ω)
31 sdomdom 8524 . . . . . . . . . . 11 ((𝑓𝑛) ≺ ω → (𝑓𝑛) ≼ ω)
3230, 31syl 17 . . . . . . . . . 10 (((𝑓𝑛) ≼ 𝑛𝑛 ≺ ω) → (𝑓𝑛) ≼ ω)
3328, 29, 32syl2anr 599 . . . . . . . . 9 ((𝑛 ∈ ω ∧ (𝑓𝑛) ≈ 𝑛) → (𝑓𝑛) ≼ ω)
3433ralimiaa 3130 . . . . . . . 8 (∀𝑛 ∈ ω (𝑓𝑛) ≈ 𝑛 → ∀𝑛 ∈ ω (𝑓𝑛) ≼ ω)
35 iunctb2 34821 . . . . . . . 8 (∀𝑛 ∈ ω (𝑓𝑛) ≼ ω → 𝑛 ∈ ω (𝑓𝑛) ≼ ω)
3627, 34, 353syl 18 . . . . . . 7 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → 𝑛 ∈ ω (𝑓𝑛) ≼ ω)
3736adantl 485 . . . . . 6 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → 𝑛 ∈ ω (𝑓𝑛) ≼ ω)
3825, 37eqbrtrd 5055 . . . . 5 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ran 𝑓 ≼ ω)
39 fvssunirn 6678 . . . . . . . . . 10 (𝑓𝑛) ⊆ ran 𝑓
4039jctl 527 . . . . . . . . 9 ((𝑓𝑛) ≈ 𝑛 → ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛))
4140adantl 485 . . . . . . . 8 (((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛))
4241ralimi 3131 . . . . . . 7 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛))
43 sseq1 3943 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑛) → (𝑥 ran 𝑓 ↔ (𝑓𝑛) ⊆ ran 𝑓))
4443, 4anbi12d 633 . . . . . . . . . . 11 (𝑥 = (𝑓𝑛) → ((𝑥 ran 𝑓𝑥𝑛) ↔ ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛)))
459, 44spcev 3558 . . . . . . . . . 10 (((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛) → ∃𝑥(𝑥 ran 𝑓𝑥𝑛))
4645ralimi 3131 . . . . . . . . 9 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ∃𝑥(𝑥 ran 𝑓𝑥𝑛))
47 isinf2 34823 . . . . . . . . 9 (∀𝑛 ∈ ω ∃𝑥(𝑥 ran 𝑓𝑥𝑛) → ¬ ran 𝑓 ∈ Fin)
4846, 47syl 17 . . . . . . . 8 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛) → ¬ ran 𝑓 ∈ Fin)
49 vex 3447 . . . . . . . . . . 11 𝑓 ∈ V
5049rnex 7603 . . . . . . . . . 10 ran 𝑓 ∈ V
5150uniex 7451 . . . . . . . . 9 ran 𝑓 ∈ V
52 infinf 9981 . . . . . . . . 9 ( ran 𝑓 ∈ V → (¬ ran 𝑓 ∈ Fin ↔ ω ≼ ran 𝑓))
5351, 52ax-mp 5 . . . . . . . 8 ran 𝑓 ∈ Fin ↔ ω ≼ ran 𝑓)
5448, 53sylib 221 . . . . . . 7 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛) → ω ≼ ran 𝑓)
5542, 54syl 17 . . . . . 6 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ω ≼ ran 𝑓)
5655adantl 485 . . . . 5 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ω ≼ ran 𝑓)
57 sbth 8625 . . . . 5 (( ran 𝑓 ≼ ω ∧ ω ≼ ran 𝑓) → ran 𝑓 ≈ ω)
5838, 56, 57syl2anc 587 . . . 4 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ran 𝑓 ≈ ω)
59 sseq1 3943 . . . . . 6 (𝑥 = ran 𝑓 → (𝑥𝐴 ran 𝑓𝐴))
60 breq1 5036 . . . . . 6 (𝑥 = ran 𝑓 → (𝑥 ≈ ω ↔ ran 𝑓 ≈ ω))
6159, 60anbi12d 633 . . . . 5 (𝑥 = ran 𝑓 → ((𝑥𝐴𝑥 ≈ ω) ↔ ( ran 𝑓𝐴 ran 𝑓 ≈ ω)))
6251, 61spcev 3558 . . . 4 (( ran 𝑓𝐴 ran 𝑓 ≈ ω) → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
6318, 58, 62syl2anc 587 . . 3 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
6463exlimiv 1931 . 2 (∃𝑓(𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
651, 6, 643syl 18 1 𝐴 ∈ Fin → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2112  wral 3109  Vcvv 3444  wss 3884  𝒫 cpw 4500   cuni 4803   ciun 4884   class class class wbr 5033  cmpt 5113  ran crn 5524   Fn wfn 6323  cfv 6328  ωcom 7564  cen 8493  cdom 8494  csdm 8495  Fincfn 8496
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-reg 9044  ax-inf2 9092  ax-cc 9850  ax-ac2 9878
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-oi 8962  df-r1 9181  df-rank 9182  df-card 9356  df-acn 9359  df-ac 9531
This theorem is referenced by: (None)
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