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Theorem ctbssinf 37389
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 9294 . 2 𝐴 ∈ Fin → ∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛))
2 omex 9681 . . 3 ω ∈ V
3 sseq1 4021 . . . 4 (𝑥 = (𝑓𝑛) → (𝑥𝐴 ↔ (𝑓𝑛) ⊆ 𝐴))
4 breq1 5151 . . . 4 (𝑥 = (𝑓𝑛) → (𝑥𝑛 ↔ (𝑓𝑛) ≈ 𝑛))
53, 4anbi12d 632 . . 3 (𝑥 = (𝑓𝑛) → ((𝑥𝐴𝑥𝑛) ↔ ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)))
62, 5ac6s2 10524 . 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 3081 . . . . 5 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓𝑛) ⊆ 𝐴)
9 fvex 6920 . . . . . . . 8 (𝑓𝑛) ∈ V
109elpw 4609 . . . . . . 7 ((𝑓𝑛) ∈ 𝒫 𝐴 ↔ (𝑓𝑛) ⊆ 𝐴)
1110ralbii 3091 . . . . . 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 4920 . . . . . . . 8 (ran 𝑓 ⊆ 𝒫 𝐴 ran 𝑓 𝒫 𝐴)
14 unipw 5461 . . . . . . . 8 𝒫 𝐴 = 𝐴
1513, 14sseqtrdi 4046 . . . . . . 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 5952 . . . . . . . . 9 (𝑓 Fn ω → ran 𝑓 = ran (𝑛 ∈ ω ↦ (𝑓𝑛)))
2221unieqd 4925 . . . . . . . 8 (𝑓 Fn ω → ran 𝑓 = ran (𝑛 ∈ ω ↦ (𝑓𝑛)))
239dfiun3 5983 . . . . . . . 8 𝑛 ∈ ω (𝑓𝑛) = ran (𝑛 ∈ ω ↦ (𝑓𝑛))
2422, 23eqtr4di 2793 . . . . . . 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 3081 . . . . . . . 8 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω (𝑓𝑛) ≈ 𝑛)
28 endom 9018 . . . . . . . . . 10 ((𝑓𝑛) ≈ 𝑛 → (𝑓𝑛) ≼ 𝑛)
29 nnsdom 9692 . . . . . . . . . 10 (𝑛 ∈ ω → 𝑛 ≺ ω)
30 domsdomtr 9151 . . . . . . . . . . 11 (((𝑓𝑛) ≼ 𝑛𝑛 ≺ ω) → (𝑓𝑛) ≺ ω)
31 sdomdom 9019 . . . . . . . . . . 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 3080 . . . . . . . 8 (∀𝑛 ∈ ω (𝑓𝑛) ≈ 𝑛 → ∀𝑛 ∈ ω (𝑓𝑛) ≼ ω)
35 iunctb2 37386 . . . . . . . 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 5170 . . . . 5 ((𝑓 Fn ω ∧ ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛)) → ran 𝑓 ≼ ω)
39 fvssunirn 6940 . . . . . . . . . 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 3081 . . . . . . 7 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ 𝐴 ∧ (𝑓𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛))
43 sseq1 4021 . . . . . . . . . . . 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 3081 . . . . . . . . 9 (∀𝑛 ∈ ω ((𝑓𝑛) ⊆ ran 𝑓 ∧ (𝑓𝑛) ≈ 𝑛) → ∀𝑛 ∈ ω ∃𝑥(𝑥 ran 𝑓𝑥𝑛))
47 isinf2 37388 . . . . . . . . 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 3482 . . . . . . . . . . 11 𝑓 ∈ V
5049rnex 7933 . . . . . . . . . 10 ran 𝑓 ∈ V
5150uniex 7760 . . . . . . . . 9 ran 𝑓 ∈ V
52 infinf 10604 . . . . . . . . 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 9132 . . . . 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 4021 . . . . . 6 (𝑥 = ran 𝑓 → (𝑥𝐴 ran 𝑓𝐴))
60 breq1 5151 . . . . . 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 1928 . 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 1537  wex 1776  wcel 2106  wral 3059  Vcvv 3478  wss 3963  𝒫 cpw 4605   cuni 4912   ciun 4996   class class class wbr 5148  cmpt 5231  ran crn 5690   Fn wfn 6558  cfv 6563  ωcom 7887  cen 8981  cdom 8982  csdm 8983  Fincfn 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679  ax-cc 10473  ax-ac2 10501
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-r1 9802  df-rank 9803  df-card 9977  df-acn 9980  df-ac 10154
This theorem is referenced by: (None)
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