Theorem elopg 5358

Description: Characterization of the elements of an ordered pair. Closed form of elop 5359. (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 4801	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
2		eleq2 2901	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ 𝐶 ∈ {{𝐴}, {𝐴, 𝐵}}))
3		snex 5332	. . . 4 ⊢ {𝐴} ∈ V
4		prex 5333	. . . 4 ⊢ {𝐴, 𝐵} ∈ V
5	3, 4	elpr2 4591	. . 3 ⊢ (𝐶 ∈ {{𝐴}, {𝐴, 𝐵}} ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵}))
6	2, 5	syl6bb 289	. 2 ⊢ (⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}} → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))
7	1, 6	syl 17	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐶 ∈ ⟨𝐴, 𝐵⟩ ↔ (𝐶 = {𝐴} ∨ 𝐶 = {𝐴, 𝐵})))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537   ∈ wcel 2114  {csn 4567  {cpr 4569  ⟨cop 4573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574

This theorem is referenced by:  elop  5359  bj-inftyexpidisj  34495