Theorem opwo0id 5472

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 5471	. . . 4 ⊢ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩
2		disjsn 4687	. . . 4 ⊢ ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩)
3	1, 2	mpbir 231	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅
4		disjdif2 4455	. . 3 ⊢ ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ → (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩)
5	3, 4	ax-mp 5	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩
6	5	eqcomi 2744	1 ⊢ ⟨𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅})

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108   ∖ cdif 3923   ∩ cin 3925  ∅c0 4308  {csn 4601  ⟨cop 4607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608

This theorem is referenced by:  fundmge2nop0  14520