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Theorem ensn1g 9019
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
Assertion
Ref Expression
ensn1g (𝐴𝑉 → {𝐴} ≈ 1o)

Proof of Theorem ensn1g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 4639 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21breq1d 5159 . 2 (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o))
3 vex 3479 . . 3 𝑥 ∈ V
43ensn1 9017 . 2 {𝑥} ≈ 1o
52, 4vtoclg 3557 1 (𝐴𝑉 → {𝐴} ≈ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {csn 4629   class class class wbr 5149  1oc1o 8459  cen 8936
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-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
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 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-suc 6371  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-1o 8466  df-en 8940
This theorem is referenced by:  enpr1g  9020  en1b  9023  en1bOLD  9024  snmapen1  9039  en2snOLDOLD  9043  snfi  9044  enpr2dOLD  9050  snnen2oOLD  9227  sucxpdom  9255  en1eqsnOLD  9275  en1eqsnbi  9276  pr2nelemOLD  9998  prdom2  10001  dju1en  10166  triv1nsgd  19053  snct  31938  rngoueqz  36808  safesnsupfidom1o  42168  sn1dom  42277
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