Theorem opideq 34110

Description: Equality conditions for ordered pairs ⟨𝐴, 𝐴⟩ and ⟨𝐵, 𝐵⟩. (Contributed by Peter Mazsa, 22-Jul-2019.)

Assertion

Ref	Expression
opideq	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ 𝐴 = 𝐵))

Proof of Theorem opideq

Step	Hyp	Ref	Expression
1		opidORIG 34109	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⟨𝐴, 𝐴⟩ = {{𝐴}})
2		opidORIG 34109	. . 3 ⊢ (𝐵 ∈ 𝑊 → ⟨𝐵, 𝐵⟩ = {{𝐵}})
3	1, 2	eqeqan12d 2638	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ {{𝐴}} = {{𝐵}}))
4		snex 4908	. . . . 5 ⊢ {𝐴} ∈ V
5		sneqbg 4374	. . . . 5 ⊢ ({𝐴} ∈ V → ({{𝐴}} = {{𝐵}} ↔ {𝐴} = {𝐵}))
6	4, 5	ax-mp 5	. . . 4 ⊢ ({{𝐴}} = {{𝐵}} ↔ {𝐴} = {𝐵})
7		sneqbg 4374	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
8	6, 7	syl5bb 272	. . 3 ⊢ (𝐴 ∈ 𝑉 → ({{𝐴}} = {{𝐵}} ↔ 𝐴 = 𝐵))
9	8	adantr 481	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ({{𝐴}} = {{𝐵}} ↔ 𝐴 = 𝐵))
10	3, 9	bitrd 268	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐴⟩ = ⟨𝐵, 𝐵⟩ ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  ⟨cop 4183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184

This theorem is referenced by: (None)