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Theorem ensn1g 7039
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 3702 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21breq1d 4121 . 2 (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o))
3 vex 2818 . . 3 𝑥 ∈ V
43ensn1 7038 . 2 {𝑥} ≈ 1o
52, 4vtoclg 2877 1 (𝐴𝑉 → {𝐴} ≈ 1o)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1398  wcel 2205  {csn 3691   class class class wbr 4111  1oc1o 6642  cen 6975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-suc 4494  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-1o 6649  df-en 6978
This theorem is referenced by:  enpr1g  7040  en1bg  7042  en2sn  7057  snfig  7058  enpr2d  7066  snnen2og  7115  eqsndc  7165  en1eqsn  7220  en1eqsnbi  7221  pr2nelem  7490  dju1en  7522  triv1nsgd  13952
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