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Theorem snen1g 42260
Description: A singleton is equinumerous to ordinal one iff its content is a set. (Contributed by RP, 8-Oct-2023.)
Assertion
Ref Expression
snen1g ({𝐴} ≈ 1o𝐴 ∈ V)

Proof of Theorem snen1g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqcom 2739 . . . 4 ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴})
2 vex 3478 . . . . . 6 𝑥 ∈ V
32sneqr 4840 . . . . 5 ({𝑥} = {𝐴} → 𝑥 = 𝐴)
4 sneq 4637 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
53, 4impbii 208 . . . 4 ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
61, 5bitri 274 . . 3 ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴)
76exbii 1850 . 2 (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴)
8 en1 9017 . 2 ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥})
9 isset 3487 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
107, 8, 93bitr4i 302 1 ({𝐴} ≈ 1o𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wex 1781  wcel 2106  Vcvv 3474  {csn 4627   class class class wbr 5147  1oc1o 8455  cen 8932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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-uni 4908  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-res 5687  df-ima 5688  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-1o 8462  df-en 8936
This theorem is referenced by:  snen1el  42261
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