Theorem opprc 3854

Description: Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
opprc	⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)

Proof of Theorem opprc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-op 3652	. 2 ⊢ ⟨𝐴, 𝐵⟩ = {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})}
2		3simpa 997	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	con3i 633	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
4	3	alrimiv 1898	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∀𝑥 ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
5		abeq0 3499	. . 3 ⊢ ({𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = ∅ ↔ ∀𝑥 ¬ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}}))
6	4, 5	sylibr 134	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {𝑥 ∣ (𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝑥 ∈ {{𝐴}, {𝐴, 𝐵}})} = ∅)
7	1, 6	eqtrid 2252	1 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∧ w3a 981  ∀wal 1371   = wceq 1373   ∈ wcel 2178  {cab 2193  Vcvv 2776  ∅c0 3468  {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-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-dif 3176  df-nul 3469  df-op 3652

This theorem is referenced by:  opprc1  3855  opprc2  3856  ovprc  6003