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Theorem ensn1g 7050
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 3705 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21breq1d 4124 . 2 (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o))
3 vex 2818 . . 3 𝑥 ∈ V
43ensn1 7049 . 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 3694   class class class wbr 4114  1oc1o 6653  cen 6986
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 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559
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 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-1o 6660  df-en 6989
This theorem is referenced by:  enpr1g  7051  en1bg  7053  en2sn  7068  snfig  7069  enpr2d  7077  snnen2og  7126  eqsndc  7176  en1eqsn  7231  en1eqsnbi  7232  pr2nelem  7501  dju1en  7533  triv1nsgd  14019
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