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Theorem snen1g 43537
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 2744 . . . 4 ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴})
2 vex 3484 . . . . . 6 𝑥 ∈ V
32sneqr 4840 . . . . 5 ({𝑥} = {𝐴} → 𝑥 = 𝐴)
4 sneq 4636 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
53, 4impbii 209 . . . 4 ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
61, 5bitri 275 . . 3 ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴)
76exbii 1848 . 2 (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴)
8 en1 9064 . 2 ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥})
9 isset 3494 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
107, 8, 93bitr4i 303 1 ({𝐴} ≈ 1o𝐴 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  {csn 4626   class class class wbr 5143  1oc1o 8499  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1o 8506  df-en 8986
This theorem is referenced by:  snen1el  43538
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