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Theorem infpss 8999
Description: Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 9095. (Contributed by NM, 23-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
infpss (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem infpss
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 infn0 8182 . . 3 (ω ≼ 𝐴𝐴 ≠ ∅)
2 n0 3913 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦𝐴)
31, 2sylib 208 . 2 (ω ≼ 𝐴 → ∃𝑦 𝑦𝐴)
4 reldom 7921 . . . . . 6 Rel ≼
54brrelex2i 5129 . . . . 5 (ω ≼ 𝐴𝐴 ∈ V)
6 difexg 4778 . . . . 5 (𝐴 ∈ V → (𝐴 ∖ {𝑦}) ∈ V)
75, 6syl 17 . . . 4 (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ∈ V)
87adantr 481 . . 3 ((ω ≼ 𝐴𝑦𝐴) → (𝐴 ∖ {𝑦}) ∈ V)
9 simpr 477 . . . . 5 ((ω ≼ 𝐴𝑦𝐴) → 𝑦𝐴)
10 difsnpss 4314 . . . . 5 (𝑦𝐴 ↔ (𝐴 ∖ {𝑦}) ⊊ 𝐴)
119, 10sylib 208 . . . 4 ((ω ≼ 𝐴𝑦𝐴) → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
12 infdifsn 8514 . . . . 5 (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ≈ 𝐴)
1312adantr 481 . . . 4 ((ω ≼ 𝐴𝑦𝐴) → (𝐴 ∖ {𝑦}) ≈ 𝐴)
1411, 13jca 554 . . 3 ((ω ≼ 𝐴𝑦𝐴) → ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴))
15 psseq1 3678 . . . . 5 (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥𝐴 ↔ (𝐴 ∖ {𝑦}) ⊊ 𝐴))
16 breq1 4626 . . . . 5 (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥𝐴 ↔ (𝐴 ∖ {𝑦}) ≈ 𝐴))
1715, 16anbi12d 746 . . . 4 (𝑥 = (𝐴 ∖ {𝑦}) → ((𝑥𝐴𝑥𝐴) ↔ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴)))
1817spcegv 3284 . . 3 ((𝐴 ∖ {𝑦}) ∈ V → (((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴) → ∃𝑥(𝑥𝐴𝑥𝐴)))
198, 14, 18sylc 65 . 2 ((ω ≼ 𝐴𝑦𝐴) → ∃𝑥(𝑥𝐴𝑥𝐴))
203, 19exlimddv 1860 1 (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  Vcvv 3190  cdif 3557  wpss 3561  c0 3897  {csn 4155   class class class wbr 4623  ωcom 7027  cen 7912  cdom 7913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-om 7028  df-1o 7520  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919
This theorem is referenced by:  isfin4-2  9096
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