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Theorem infpss 9631
 Description: Every infinite set has an equinumerous proper subset, proved without AC or Infinity. Exercise 7 of [TakeutiZaring] p. 91. See also infpssALT 9727. (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 8772 . . 3 (ω ≼ 𝐴𝐴 ≠ ∅)
2 n0 4308 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦𝐴)
31, 2sylib 220 . 2 (ω ≼ 𝐴 → ∃𝑦 𝑦𝐴)
4 reldom 8507 . . . . . 6 Rel ≼
54brrelex2i 5602 . . . . 5 (ω ≼ 𝐴𝐴 ∈ V)
6 difexg 5222 . . . . 5 (𝐴 ∈ V → (𝐴 ∖ {𝑦}) ∈ V)
75, 6syl 17 . . . 4 (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ∈ V)
87adantr 483 . . 3 ((ω ≼ 𝐴𝑦𝐴) → (𝐴 ∖ {𝑦}) ∈ V)
9 simpr 487 . . . . 5 ((ω ≼ 𝐴𝑦𝐴) → 𝑦𝐴)
10 difsnpss 4732 . . . . 5 (𝑦𝐴 ↔ (𝐴 ∖ {𝑦}) ⊊ 𝐴)
119, 10sylib 220 . . . 4 ((ω ≼ 𝐴𝑦𝐴) → (𝐴 ∖ {𝑦}) ⊊ 𝐴)
12 infdifsn 9112 . . . . 5 (ω ≼ 𝐴 → (𝐴 ∖ {𝑦}) ≈ 𝐴)
1312adantr 483 . . . 4 ((ω ≼ 𝐴𝑦𝐴) → (𝐴 ∖ {𝑦}) ≈ 𝐴)
1411, 13jca 514 . . 3 ((ω ≼ 𝐴𝑦𝐴) → ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴))
15 psseq1 4062 . . . 4 (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥𝐴 ↔ (𝐴 ∖ {𝑦}) ⊊ 𝐴))
16 breq1 5060 . . . 4 (𝑥 = (𝐴 ∖ {𝑦}) → (𝑥𝐴 ↔ (𝐴 ∖ {𝑦}) ≈ 𝐴))
1715, 16anbi12d 632 . . 3 (𝑥 = (𝐴 ∖ {𝑦}) → ((𝑥𝐴𝑥𝐴) ↔ ((𝐴 ∖ {𝑦}) ⊊ 𝐴 ∧ (𝐴 ∖ {𝑦}) ≈ 𝐴)))
188, 14, 17spcedv 3597 . 2 ((ω ≼ 𝐴𝑦𝐴) → ∃𝑥(𝑥𝐴𝑥𝐴))
193, 18exlimddv 1929 1 (ω ≼ 𝐴 → ∃𝑥(𝑥𝐴𝑥𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1530  ∃wex 1773   ∈ wcel 2107   ≠ wne 3014  Vcvv 3493   ∖ cdif 3931   ⊊ wpss 3935  ∅c0 4289  {csn 4559   class class class wbr 5057  ωcom 7572   ≈ cen 8498   ≼ cdom 8499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7573  df-1o 8094  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505 This theorem is referenced by:  isfin4-2  9728
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