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Theorem opwo0id 5378
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 5377 . . . 4 ¬ ∅ ∈ ⟨𝑋, 𝑌
2 disjsn 4639 . . . 4 ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩)
31, 2mpbir 232 . . 3 (⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅
4 disjdif2 4424 . . 3 ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ → (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩)
53, 4ax-mp 5 . 2 (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌
65eqcomi 2827 1 𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1528  wcel 2105  cdif 3930  cin 3932  c0 4288  {csn 4557  cop 4563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564
This theorem is referenced by:  fundmge2nop0  13838
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