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Theorem opwo0id 4951
Description: An ordered pair is equal to the ordered pair without the empty set. This is because no ordered pair contains the empty set. (Contributed by AV, 15-Nov-2021.)
Assertion
Ref Expression
opwo0id 𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅})

Proof of Theorem opwo0id
StepHypRef Expression
1 0nelop 4950 . . . 4 ¬ ∅ ∈ ⟨𝑋, 𝑌
2 disjsn 4237 . . . 4 ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩)
31, 2mpbir 221 . . 3 (⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅
4 disjdif2 4038 . . 3 ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ → (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩)
53, 4ax-mp 5 . 2 (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌
65eqcomi 2629 1 𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1481  wcel 1988  cdif 3564  cin 3566  c0 3907  {csn 4168  cop 4174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175
This theorem is referenced by:  fundmge2nop0  13257
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