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Theorem oprcl 3693
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.
StepHypRef Expression
1 elex2 2671 . 2 (𝐶 ∈ ⟨𝐴, 𝐵⟩ → ∃𝑦 𝑦 ∈ ⟨𝐴, 𝐵⟩)
2 df-op 3500 . . . . . . 7 𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
32eleq2i 2179 . . . . . 6 (𝑦 ∈ ⟨𝐴, 𝐵⟩ ↔ 𝑦 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})})
4 df-clab 2100 . . . . . 6 (𝑦 ∈ {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} ↔ [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
53, 4bitri 183 . . . . 5 (𝑦 ∈ ⟨𝐴, 𝐵⟩ ↔ [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
6 3simpa 959 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
76sbimi 1718 . . . . 5 ([𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V))
85, 7sylbi 120 . . . 4 (𝑦 ∈ ⟨𝐴, 𝐵⟩ → [𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V))
9 nfv 1489 . . . . 5 𝑥(𝐴 ∈ V ∧ 𝐵 ∈ V)
109sbf 1731 . . . 4 ([𝑦 / 𝑥](𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V))
118, 10sylib 121 . . 3 (𝑦 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
1211exlimiv 1558 . 2 (∃𝑦 𝑦 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
131, 12syl 14 1 (𝐶 ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 943  wex 1449  wcel 1461  [wsb 1716  {cab 2099  Vcvv 2655  {csn 3491  {cpr 3492  cop 3494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1404  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-3an 945  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-v 2657  df-op 3500
This theorem is referenced by:  opth1  4116  opth  4117  0nelop  4128
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