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Theorem opwo0id 5507
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 5506 . . . 4 ¬ ∅ ∈ ⟨𝑋, 𝑌
2 disjsn 4716 . . . 4 ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩)
31, 2mpbir 231 . . 3 (⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅
4 disjdif2 4486 . . 3 ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ → (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩)
53, 4ax-mp 5 . 2 (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌
65eqcomi 2744 1 𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  cdif 3960  cin 3962  c0 4339  {csn 4631  cop 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638
This theorem is referenced by:  fundmge2nop0  14538
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