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Theorem snen1g 42832
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 2733 . . . 4 ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴})
2 vex 3472 . . . . . 6 𝑥 ∈ V
32sneqr 4836 . . . . 5 ({𝑥} = {𝐴} → 𝑥 = 𝐴)
4 sneq 4633 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
53, 4impbii 208 . . . 4 ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
61, 5bitri 275 . . 3 ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴)
76exbii 1842 . 2 (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴)
8 en1 9020 . 2 ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥})
9 isset 3481 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
107, 8, 93bitr4i 303 1 ({𝐴} ≈ 1o𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wex 1773  wcel 2098  Vcvv 3468  {csn 4623   class class class wbr 5141  1oc1o 8457  cen 8935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1o 8464  df-en 8939
This theorem is referenced by:  snen1el  42833
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