Theorem sdomnen 8138

Description: Strict dominance implies non-equinumerosity. (Contributed by NM, 10-Jun-1998.)

Assertion

Ref	Expression
sdomnen	⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)

Proof of Theorem sdomnen

Step	Hyp	Ref	Expression
1		brsdom 8132	. 2 ⊢ (𝐴 ≺ 𝐵 ↔ (𝐴 ≼ 𝐵 ∧ ¬ 𝐴 ≈ 𝐵))
2	1	simprbi 484	1 ⊢ (𝐴 ≺ 𝐵 → ¬ 𝐴 ≈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   class class class wbr 4786   ≈ cen 8106   ≼ cdom 8107   ≺ csdm 8108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-br 4787  df-sdom 8112

This theorem is referenced by:  bren2  8140  domdifsn  8199  sdomnsym  8241  domnsym  8242  sdomirr  8253  php5  8304  sucdom2  8312  pssinf  8326  f1finf1o  8343  isfinite2  8374  cardom  9012  pm54.43  9026  pr2ne  9028  alephdom  9104  cdainflem  9215  ackbij1b  9263  isfin4-3  9339  fin23lem25  9348  fin67  9419  axcclem  9481  canthp1lem2  9677  gchinf  9681  pwfseqlem4  9686  tskssel  9781  1nprm  15599  en2top  21010  rp-isfinite6  38390