Theorem oprcl 3857

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 2793	. 2 ⊢ (𝐶 ∈ ⟨𝐴, 𝐵⟩ → ∃𝑦 𝑦 ∈ ⟨𝐴, 𝐵⟩)
2		df-op 3652	. . . . . . 7 ⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
3	2	eleq2i 2274	. . . . . 6 ⊢ (𝑦 ∈ ⟨𝐴, 𝐵⟩ ↔ 𝑦 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})})
4		df-clab 2194	. . . . . 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 184	. . . . 5 ⊢ (𝑦 ∈ ⟨𝐴, 𝐵⟩ ↔ [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
6		3simpa 997	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7	6	sbimi 1788	. . . . 5 ⊢ ([𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V))
8	5, 7	sylbi 121	. . . 4 ⊢ (𝑦 ∈ ⟨𝐴, 𝐵⟩ → [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V))
9		nfv 1552	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V)
10	9	sbf 1801	. . . 4 ⊢ ([𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
11	8, 10	sylib 122	. . 3 ⊢ (𝑦 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
12	11	exlimiv 1622	. 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 104   ∧ w3a 981  ∃wex 1516  [wsb 1786   ∈ wcel 2178  {cab 2193  Vcvv 2776  {csn 3643  {cpr 3644  ⟨cop 3646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-v 2778  df-op 3652

This theorem is referenced by:  opth1  4298  opth  4299  0nelop  4310