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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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