Theorem oprcl 3789

Description: If an ordered pair has an element, then its arguments are sets. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
oprcl	⊢ (𝐶 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Proof of Theorem oprcl

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex2 2746	. 2 ⊢ (𝐶 ∈ ⟨𝐴, 𝐵⟩ → ∃𝑦 𝑦 ∈ ⟨𝐴, 𝐵⟩)
2		df-op 3592	. . . . . . 7 ⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
3	2	eleq2i 2237	. . . . . 6 ⊢ (𝑦 ∈ ⟨𝐴, 𝐵⟩ ↔ 𝑦 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})})
4		df-clab 2157	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} ↔ [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
5	3, 4	bitri 183	. . . . 5 ⊢ (𝑦 ∈ ⟨𝐴, 𝐵⟩ ↔ [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
6		3simpa 989	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7	6	sbimi 1757	. . . . 5 ⊢ ([𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V))
8	5, 7	sylbi 120	. . . 4 ⊢ (𝑦 ∈ ⟨𝐴, 𝐵⟩ → [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V))
9		nfv 1521	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V)
10	9	sbf 1770	. . . 4 ⊢ ([𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
11	8, 10	sylib 121	. . 3 ⊢ (𝑦 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
12	11	exlimiv 1591	. 2 ⊢ (∃𝑦 𝑦 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
13	1, 12	syl 14	1 ⊢ (𝐶 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 973  ∃wex 1485  [wsb 1755   ∈ wcel 2141  {cab 2156  Vcvv 2730  {csn 3583  {cpr 3584  ⟨cop 3586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-op 3592

This theorem is referenced by:  opth1  4221  opth  4222  0nelop  4233