Theorem opwo0id 5469

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 5468	. . . 4 ⊢ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩
2		disjsn 4672	. . . 4 ⊢ ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ ⟨𝑋, 𝑌⟩)
3	1, 2	mpbir 234	. . 3 ⊢ (⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅
4		disjdif2 4436	. . 3 ⊢ ((⟨𝑋, 𝑌⟩ ∩ {∅}) = ∅ → (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩)
5	3, 4	ax-mp 5	. 2 ⊢ (⟨𝑋, 𝑌⟩ ∖ {∅}) = ⟨𝑋, 𝑌⟩
6	5	eqcomi 2770	1 ⊢ ⟨𝑋, 𝑌⟩ = (⟨𝑋, 𝑌⟩ ∖ {∅})

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2145   ∖ cdif 3896   ∩ cin 3898  ∅c0 4279  {csn 4584  ⟨cop 4590

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591

This theorem is used by:  fundmge2nop0  14627