Theorem opwo0id 5471

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 5470	. . . 4 ⊢ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩
2		disjsn 4673	. . . 4 ⊢ ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩)
3	1, 2	mpbir 234	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅
4		disjdif2 4437	. . 3 ⊢ ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ → (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩)
5	3, 4	ax-mp 5	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩
6	5	eqcomi 2774	1 ⊢ ⟨𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅})

Syntax hints:  ¬ wn 3   = wceq 1563   ∈ wcel 2145   ∖ cdif 3904   ∩ cin 3906  ∅c0 4288  {csn 4585  ⟨cop 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592

This theorem is referenced by:  fundmge2nop0  14529