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Theorem ctbssinf 37407
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 9296 . 2 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛))
2 omex 9683 . . 3 ω ∈ V
3 sseq1 4009 . . . 4 (𝑥 = (𝑓𝑛) → (𝑥𝐴 ↔ (𝑓𝑛) ⊆ 𝐴))
4 breq1 5146 . . . 4 (𝑥 = (𝑓𝑛) → (𝑥𝑛 ↔ (𝑓𝑛) ≈ 𝑛))
53, 4anbi12d 632 . . 3 (𝑥 = (𝑓𝑛) → ((𝑥𝐴𝑥𝑛) ↔ ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)))
62, 5ac6s2 10526 . 2 (∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛) → ∃𝑓(𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)))
7 simpl 482 . . . . . 6 (((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → (𝑓𝑛) ⊆ 𝐴)
87ralimi 3083 . . . . 5 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓𝑛) ⊆ 𝐴)
9 fvex 6919 . . . . . . . 8 (𝑓𝑛) ∈ V
109elpw 4604 . . . . . . 7 ((𝑓𝑛) ∈ 𝒫 𝐴 ↔ (𝑓𝑛) ⊆ 𝐴)
1110ralbii 3093 . . . . . 6 (∀𝑛 ∈ ω (𝑓𝑛) ∈ 𝒫 𝐴 ↔ ∀𝑛 ∈ ω (𝑓𝑛) ⊆ 𝐴)
12 fnfvrnss 7141 . . . . . . 7 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓𝑛) ∈ 𝒫 𝐴) → ran 𝑓 ⊆ 𝒫 𝐴)
13 uniss 4915 . . . . . . . 8 (ran 𝑓 ⊆ 𝒫 𝐴 ran 𝑓 𝒫 𝐴)
14 unipw 5455 . . . . . . . 8 𝒫 𝐴 = 𝐴
1513, 14sseqtrdi 4024 . . . . . . 7 (ran 𝑓 ⊆ 𝒫 𝐴 ran 𝑓𝐴)
1612, 15syl 17 . . . . . 6 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓𝑛) ∈ 𝒫 𝐴) → ran 𝑓𝐴)
1711, 16sylan2br 595 . . . . 5 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω (𝑓𝑛) ⊆ 𝐴) → ran 𝑓𝐴)
188, 17sylan2 593 . . . 4 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ran 𝑓𝐴)
19 dffn5 6967 . . . . . . . . . . 11 (𝑓 Fn ω ↔ 𝑓 = (𝑛 ∈ ω ↦ (𝑓𝑛)))
2019biimpi 216 . . . . . . . . . 10 (𝑓 Fn ω → 𝑓 = (𝑛 ∈ ω ↦ (𝑓𝑛)))
2120rneqd 5949 . . . . . . . . 9 (𝑓 Fn ω → ran 𝑓 = ran (𝑛 ∈ ω ↦ (𝑓𝑛)))
2221unieqd 4920 . . . . . . . 8 (𝑓 Fn ω → ran 𝑓 = ran (𝑛 ∈ ω ↦ (𝑓𝑛)))
239dfiun3 5980 . . . . . . . 8 𝑛 ∈ ω (𝑓𝑛) = ran (𝑛 ∈ ω ↦ (𝑓𝑛))
2422, 23eqtr4di 2795 . . . . . . 7 (𝑓 Fn ω → ran 𝑓 = 𝑛 ∈ ω (𝑓𝑛))
2524adantr 480 . . . . . 6 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ran 𝑓 = 𝑛 ∈ ω (𝑓𝑛))
26 simpr 484 . . . . . . . . 9 (((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → (𝑓𝑛) ≈ 𝑛)
2726ralimi 3083 . . . . . . . 8 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓𝑛) ≈ 𝑛)
28 endom 9019 . . . . . . . . . 10 ((𝑓𝑛) ≈ 𝑛 → (𝑓𝑛) ≼ 𝑛)
29 nnsdom 9694 . . . . . . . . . 10 (𝑛 ∈ ω → 𝑛 ≺ ω)
30 domsdomtr 9152 . . . . . . . . . . 11 (((𝑓𝑛) ≼ 𝑛𝑛 ≺ ω) → (𝑓𝑛) ≺ ω)
31 sdomdom 9020 . . . . . . . . . . 11 ((𝑓𝑛) ≺ ω → (𝑓𝑛) ≼ ω)
3230, 31syl 17 . . . . . . . . . 10 (((𝑓𝑛) ≼ 𝑛𝑛 ≺ ω) → (𝑓𝑛) ≼ ω)
3328, 29, 32syl2anr 597 . . . . . . . . 9 ((𝑛 ∈ ω ∧ (𝑓𝑛) ≈ 𝑛) → (𝑓𝑛) ≼ ω)
3433ralimiaa 3082 . . . . . . . 8 (∀𝑛 ∈ ω (𝑓𝑛) ≈ 𝑛 → ∀𝑛 ∈ ω (𝑓𝑛) ≼ ω)
35 iunctb2 37404 . . . . . . . 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 481 . . . . . 6 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → 𝑛 ∈ ω (𝑓𝑛) ≼ ω)
3825, 37eqbrtrd 5165 . . . . 5 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ran 𝑓 ≼ ω)
39 fvssunirn 6939 . . . . . . . . . 10 (𝑓𝑛) ⊆ ran 𝑓
4039jctl 523 . . . . . . . . 9 ((𝑓𝑛) ≈ 𝑛 → ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛))
4140adantl 481 . . . . . . . 8 (((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛))
4241ralimi 3083 . . . . . . 7 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛))
43 sseq1 4009 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑛) → (𝑥 ran 𝑓 ↔ (𝑓𝑛) ⊆ ran 𝑓))
4443, 4anbi12d 632 . . . . . . . . . . 11 (𝑥 = (𝑓𝑛) → ((𝑥 ran 𝑓𝑥𝑛) ↔ ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛)))
459, 44spcev 3606 . . . . . . . . . 10 (((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛) → ∃𝑥(𝑥 ran 𝑓𝑥𝑛))
4645ralimi 3083 . . . . . . . . 9 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ∃𝑥(𝑥 ran 𝑓𝑥𝑛))
47 isinf2 37406 . . . . . . . . 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 3484 . . . . . . . . . . 11 𝑓 ∈ V
5049rnex 7932 . . . . . . . . . 10 ran 𝑓 ∈ V
5150uniex 7761 . . . . . . . . 9 ran 𝑓 ∈ V
52 infinf 10606 . . . . . . . . 9 ( ran 𝑓 ∈ V → (¬ ran 𝑓 ∈ Fin ↔ ω ≼ ran 𝑓))
5351, 52ax-mp 5 . . . . . . . 8 ran 𝑓 ∈ Fin ↔ ω ≼ ran 𝑓)
5448, 53sylib 218 . . . . . . 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 481 . . . . 5 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ω ≼ ran 𝑓)
57 sbth 9133 . . . . 5 (( ran 𝑓 ≼ ω ∧ ω ≼ ran 𝑓) → ran 𝑓 ≈ ω)
5838, 56, 57syl2anc 584 . . . 4 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ran 𝑓 ≈ ω)
59 sseq1 4009 . . . . . 6 (𝑥 = ran 𝑓 → (𝑥𝐴 ran 𝑓𝐴))
60 breq1 5146 . . . . . 6 (𝑥 = ran 𝑓 → (𝑥 ≈ ω ↔ ran 𝑓 ≈ ω))
6159, 60anbi12d 632 . . . . 5 (𝑥 = ran 𝑓 → ((𝑥𝐴𝑥 ≈ ω) ↔ ( ran 𝑓𝐴 ran 𝑓 ≈ ω)))
6251, 61spcev 3606 . . . 4 (( ran 𝑓𝐴 ran 𝑓 ≈ ω) → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
6318, 58, 62syl2anc 584 . . 3 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ∃𝑥(𝑥𝐴𝑥 ≈ ω))
6463exlimiv 1930 . 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 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  Vcvv 3480  wss 3951  𝒫 cpw 4600   cuni 4907   ciun 4991   class class class wbr 5143  cmpt 5225  ran crn 5686   Fn wfn 6556  cfv 6561  ωcom 7887  cen 8982  cdom 8983  csdm 8984  Fincfn 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681  ax-cc 10475  ax-ac2 10503
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-r1 9804  df-rank 9805  df-card 9979  df-acn 9982  df-ac 10156
This theorem is referenced by: (None)
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