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Theorem ensn1g 9007
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 4634 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21breq1d 5154 . 2 (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o))
3 vex 3479 . . 3 𝑥 ∈ V
43ensn1 9005 . 2 {𝑥} ≈ 1o
52, 4vtoclg 3555 1 (𝐴𝑉 → {𝐴} ≈ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  {csn 4624   class class class wbr 5144  1oc1o 8446  cen 8924
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 5295  ax-nul 5302  ax-pr 5423
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 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-suc 6362  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-1o 8453  df-en 8928
This theorem is referenced by:  enpr1g  9008  en1b  9011  en1bOLD  9012  snmapen1  9027  en2snOLDOLD  9031  snfi  9032  enpr2dOLD  9038  snnen2oOLD  9215  sucxpdom  9243  en1eqsnOLD  9263  en1eqsnbi  9264  pr2nelemOLD  9985  prdom2  9988  dju1en  10153  triv1nsgd  19038  snct  31909  rngoueqz  36714  safesnsupfidom1o  42039  sn1dom  42148
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