Theorem opwo0id 5411

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 5410	. . . 4 ⊢ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩
2		disjsn 4647	. . . 4 ⊢ ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩)
3	1, 2	mpbir 230	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅
4		disjdif2 4413	. . 3 ⊢ ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ → (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩)
5	3, 4	ax-mp 5	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩
6	5	eqcomi 2747	1 ⊢ ⟨𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅})

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2106   ∖ cdif 3884   ∩ cin 3886  ∅c0 4256  {csn 4561  ⟨cop 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568

This theorem is referenced by:  fundmge2nop0  14206