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Theorem snen1el 42958
Description: A singleton is equinumerous to ordinal one if its content is an element of it. (Contributed by RP, 8-Oct-2023.)
Assertion
Ref Expression
snen1el ({𝐴} ≈ 1o𝐴 ∈ {𝐴})

Proof of Theorem snen1el
StepHypRef Expression
1 snen1g 42957 . 2 ({𝐴} ≈ 1o𝐴 ∈ V)
2 snidb 4666 . 2 (𝐴 ∈ V ↔ 𝐴 ∈ {𝐴})
31, 2bitri 274 1 ({𝐴} ≈ 1o𝐴 ∈ {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2098  Vcvv 3471  {csn 4630   class class class wbr 5150  1oc1o 8484  cen 8965
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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-1o 8491  df-en 8969
This theorem is referenced by: (None)
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