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Theorem opwo0id 5495
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 5494 . . . 4 ¬ ∅ ∈ ⟨𝑋, 𝑌
2 disjsn 4710 . . . 4 ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩)
31, 2mpbir 230 . . 3 (⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅
4 disjdif2 4474 . . 3 ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ → (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩)
53, 4ax-mp 5 . 2 (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌
65eqcomi 2735 1 𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  wcel 2099  cdif 3943  cin 3945  c0 4322  {csn 4623  cop 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630
This theorem is referenced by:  fundmge2nop0  14506
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