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Theorem opwo0id 5237
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 5236 . . . 4 ¬ ∅ ∈ ⟨𝑋, 𝑌
2 disjsn 4515 . . . 4 ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩)
31, 2mpbir 223 . . 3 (⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅
4 disjdif2 4305 . . 3 ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ → (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩)
53, 4ax-mp 5 . 2 (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌
65eqcomi 2781 1 𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1507  wcel 2050  cdif 3820  cin 3822  c0 4172  {csn 4435  cop 4441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442
This theorem is referenced by:  fundmge2nop0  13655
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