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Theorem ensucne0 41808
Description: A class equinumerous to a successor is never empty. (Contributed by RP, 11-Nov-2023.) (Proof shortened by SN, 16-Nov-2023.)
Assertion
Ref Expression
ensucne0 (𝐴 ≈ suc 𝐵𝐴 ≠ ∅)

Proof of Theorem ensucne0
StepHypRef Expression
1 nsuceq0 6401 . . . 4 suc 𝐵 ≠ ∅
2 en0r 8961 . . . 4 (∅ ≈ suc 𝐵 ↔ suc 𝐵 = ∅)
31, 2nemtbir 3041 . . 3 ¬ ∅ ≈ suc 𝐵
4 breq1 5109 . . 3 (𝐴 = ∅ → (𝐴 ≈ suc 𝐵 ↔ ∅ ≈ suc 𝐵))
53, 4mtbiri 327 . 2 (𝐴 = ∅ → ¬ 𝐴 ≈ suc 𝐵)
65necon2ai 2974 1 (𝐴 ≈ suc 𝐵𝐴 ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wne 2944  c0 4283   class class class wbr 5106  suc csuc 6320  cen 8881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-suc 6324  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-en 8885
This theorem is referenced by: (None)
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