Theorem elopg 5434

Description: Characterization of the elements of an ordered pair. Closed form of elop 5435. (Contributed by BJ, 22-Jun-2019.) (Avoid depending on this detail.)

Assertion

Ref	Expression
elopg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))

Proof of Theorem elopg

Step	Hyp	Ref	Expression
1		dfopg 4837	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2		eleq2 2849	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ 𝐶 ∈ {{𝐴}, {𝐴, 𝐵}}))
3		snex 5396	. . . 4 ⊢ {𝐴} ∈ V
4		prex 5395	. . . 4 ⊢ {𝐴, 𝐵} ∈ V
5	3, 4	elpr2 4607	. . 3 ⊢ (𝐶 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵}))
6	2, 5	bitrdi 290	. 2 ⊢ (⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))
7	1, 6	syl 18	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144  {csn 4584  {cpr 4586  ⟨cop 4834

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835

This theorem is used by:  elop  5435  bj-inftyexpidisj  38224