Theorem opwo0id 5482

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

Step	Hyp	Ref	Expression
1		0nelop 5481	. . . 4 ⊢ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩
2		disjsn 4678	. . . 4 ⊢ ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩)
3	1, 2	mpbir 234	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅
4		disjdif2 4442	. . 3 ⊢ ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ → (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩)
5	3, 4	ax-mp 5	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩
6	5	eqcomi 2772	1 ⊢ ⟨𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅})

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143   ∖ cdif 3903   ∩ cin 3905  ∅c0 4287  {csn 4590  ⟨cop 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597

This theorem is referenced by:  fundmge2nop0  14541