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Theorem snen1g 40275
 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 2805 . . . 4 ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴})
2 vex 3444 . . . . . 6 𝑥 ∈ V
32sneqr 4731 . . . . 5 ({𝑥} = {𝐴} → 𝑥 = 𝐴)
4 sneq 4535 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
53, 4impbii 212 . . . 4 ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
61, 5bitri 278 . . 3 ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴)
76exbii 1849 . 2 (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴)
8 en1 8562 . 2 ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥})
9 isset 3453 . 2 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
107, 8, 93bitr4i 306 1 ({𝐴} ≈ 1o𝐴 ∈ V)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441  {csn 4525   class class class wbr 5031  1oc1o 8081   ≈ cen 8492 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-1o 8088  df-en 8496 This theorem is referenced by:  snen1el  40276
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