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Theorem snen1g 40816
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 3412 . . . . . 6 𝑥 ∈ V
32sneqr 4751 . . . . 5 ({𝑥} = {𝐴} → 𝑥 = 𝐴)
4 sneq 4551 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
53, 4impbii 212 . . . 4 ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴)
61, 5bitri 278 . . 3 ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴)
76exbii 1855 . 2 (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴)
8 en1 8698 . 2 ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥})
9 isset 3421 . 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 1543  wex 1787  wcel 2110  Vcvv 3408  {csn 4541   class class class wbr 5053  1oc1o 8195  cen 8623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-1o 8202  df-en 8627
This theorem is referenced by:  snen1el  40817
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