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Theorem isinf2 36589
Description: The converse of isinf 9262. Any set that is not finite is literally infinite, in the sense that it contains subsets of arbitrarily large finite cardinality. (It cannot be proven that the set has countably infinite subsets unless AC is invoked.) The proof does not require the Axiom of Infinity. (Contributed by ML, 14-Dec-2020.)
Assertion
Ref Expression
isinf2 (∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛) → ¬ 𝐴 ∈ Fin)
Distinct variable group:   𝐴,𝑛,𝑥

Proof of Theorem isinf2
StepHypRef Expression
1 ssdomg 8998 . . . . . . . . 9 (𝐴 ∈ V → (𝑥𝐴𝑥𝐴))
21adantr 481 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑥𝑛) → (𝑥𝐴𝑥𝐴))
3 domen1 9121 . . . . . . . . 9 (𝑥𝑛 → (𝑥𝐴𝑛𝐴))
43adantl 482 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑥𝑛) → (𝑥𝐴𝑛𝐴))
52, 4sylibd 238 . . . . . . 7 ((𝐴 ∈ V ∧ 𝑥𝑛) → (𝑥𝐴𝑛𝐴))
65expimpd 454 . . . . . 6 (𝐴 ∈ V → ((𝑥𝑛𝑥𝐴) → 𝑛𝐴))
76ancomsd 466 . . . . 5 (𝐴 ∈ V → ((𝑥𝐴𝑥𝑛) → 𝑛𝐴))
87exlimdv 1936 . . . 4 (𝐴 ∈ V → (∃𝑥(𝑥𝐴𝑥𝑛) → 𝑛𝐴))
98ralimdv 3169 . . 3 (𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛) → ∀𝑛 ∈ ω 𝑛𝐴))
10 domalom 36588 . . 3 (∀𝑛 ∈ ω 𝑛𝐴 → ¬ 𝐴 ∈ Fin)
119, 10syl6 35 . 2 (𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛) → ¬ 𝐴 ∈ Fin))
12 prcnel 3497 . . 3 𝐴 ∈ V → ¬ 𝐴 ∈ Fin)
1312a1d 25 . 2 𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛) → ¬ 𝐴 ∈ Fin))
1411, 13pm2.61i 182 1 (∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛) → ¬ 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wex 1781  wcel 2106  wral 3061  Vcvv 3474  wss 3948   class class class wbr 5148  ωcom 7857  cen 8938  cdom 8939  Fincfn 8941
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945
This theorem is referenced by:  ctbssinf  36590
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