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Theorem isinf2 34817
Description: The converse of isinf 8719. 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 8542 . . . . . . . . 9 (𝐴 ∈ V → (𝑥𝐴𝑥𝐴))
21adantr 484 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑥𝑛) → (𝑥𝐴𝑥𝐴))
3 domen1 8647 . . . . . . . . 9 (𝑥𝑛 → (𝑥𝐴𝑛𝐴))
43adantl 485 . . . . . . . 8 ((𝐴 ∈ V ∧ 𝑥𝑛) → (𝑥𝐴𝑛𝐴))
52, 4sylibd 242 . . . . . . 7 ((𝐴 ∈ V ∧ 𝑥𝑛) → (𝑥𝐴𝑛𝐴))
65expimpd 457 . . . . . 6 (𝐴 ∈ V → ((𝑥𝑛𝑥𝐴) → 𝑛𝐴))
76ancomsd 469 . . . . 5 (𝐴 ∈ V → ((𝑥𝐴𝑥𝑛) → 𝑛𝐴))
87exlimdv 1934 . . . 4 (𝐴 ∈ V → (∃𝑥(𝑥𝐴𝑥𝑛) → 𝑛𝐴))
98ralimdv 3148 . . 3 (𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛) → ∀𝑛 ∈ ω 𝑛𝐴))
10 domalom 34816 . . 3 (∀𝑛 ∈ ω 𝑛𝐴 → ¬ 𝐴 ∈ Fin)
119, 10syl6 35 . 2 (𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛) → ¬ 𝐴 ∈ Fin))
12 prcnel 3468 . . 3 𝐴 ∈ V → ¬ 𝐴 ∈ Fin)
1312a1d 25 . 2 𝐴 ∈ V → (∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛) → ¬ 𝐴 ∈ Fin))
1411, 13pm2.61i 185 1 (∀𝑛 ∈ ω ∃𝑥(𝑥𝐴𝑥𝑛) → ¬ 𝐴 ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wex 1781  wcel 2112  wral 3109  Vcvv 3444  wss 3884   class class class wbr 5033  ωcom 7564  cen 8493  cdom 8494  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-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
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-rab 3118  df-v 3446  df-sbc 3724  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-br 5034  df-opab 5096  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  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-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-om 7565  df-1o 8089  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500
This theorem is referenced by:  ctbssinf  34818
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