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Theorem ensn1g 9021
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 4637 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21breq1d 5157 . 2 (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o))
3 vex 3476 . . 3 𝑥 ∈ V
43ensn1 9019 . 2 {𝑥} ≈ 1o
52, 4vtoclg 3541 1 (𝐴𝑉 → {𝐴} ≈ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2104  {csn 4627   class class class wbr 5147  1oc1o 8461  cen 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-mo 2532  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-suc 6369  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-1o 8468  df-en 8942
This theorem is referenced by:  enpr1g  9022  en1b  9025  en1bOLD  9026  snmapen1  9041  en2snOLDOLD  9045  snfi  9046  enpr2dOLD  9052  snnen2oOLD  9229  sucxpdom  9257  en1eqsnOLD  9277  en1eqsnbi  9278  pr2nelemOLD  10000  prdom2  10003  dju1en  10168  triv1nsgd  19089  snct  32205  rngoueqz  37111  safesnsupfidom1o  42470  sn1dom  42579
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