Theorem opelxpex2 3286

Description: The second member of an ordered pair of classes in a cross product exists when the order pair doesn't belong to I.

Assertion

Ref	Expression
opelxpex2	⊢ (⟨A, B⟩ ∈ ((C × D) ∖ I) → B ∈ V)

Proof of Theorem opelxpex2

Step	Hyp	Ref	Expression
1		eldif 2061	. 2 ⊢ (⟨A, B⟩ ∈ ((C × D) ∖ I) ↔ (⟨A, B⟩ ∈ (C × D) ⋀ ¬ ⟨A, B⟩ ∈ I))
2		opelxp1 3212	. . . . . 6 ⊢ (⟨A, B⟩ ∈ (C × D) → A ∈ C)
3		eleq1 1537	. . . . . . 7 ⊢ (⟨A, B⟩ = ⟨A, A⟩ → (⟨A, B⟩ ∈ I ↔ ⟨A, A⟩ ∈ I))
4		ididg 3285	. . . . . . . 8 ⊢ (A ∈ C → AIA)
5		df-br 2626	. . . . . . . 8 ⊢ (AIA ↔ ⟨A, A⟩ ∈ I)
6	4, 5	sylib 198	. . . . . . 7 ⊢ (A ∈ C → ⟨A, A⟩ ∈ I)
7	3, 6	syl5cbir 211	. . . . . 6 ⊢ (A ∈ C → (⟨A, B⟩ = ⟨A, A⟩ → ⟨A, B⟩ ∈ I))
8	2, 7	syl 10	. . . . 5 ⊢ (⟨A, B⟩ ∈ (C × D) → (⟨A, B⟩ = ⟨A, A⟩ → ⟨A, B⟩ ∈ I))
9		opprc2 2504	. . . . 5 ⊢ (¬ B ∈ V → ⟨A, B⟩ = ⟨A, A⟩)
10	8, 9	syl5 21	. . . 4 ⊢ (⟨A, B⟩ ∈ (C × D) → (¬ B ∈ V → ⟨A, B⟩ ∈ I))
11	10	con1d 93	. . 3 ⊢ (⟨A, B⟩ ∈ (C × D) → (¬ ⟨A, B⟩ ∈ I → B ∈ V))
12	11	imp 350	. 2 ⊢ ((⟨A, B⟩ ∈ (C × D) ⋀ ¬ ⟨A, B⟩ ∈ I) → B ∈ V)
13	1, 12	sylbi 199	1 ⊢ (⟨A, B⟩ ∈ ((C × D) ∖ I) → B ∈ V)

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  Vcvv 1814   ∖ cdif 2048  ⟨cop 2416   class class class wbr 2625  Icid 2838   × cxp 3175

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2709  ax-pow 2749  ax-pr 2786

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-br 2626  df-opab 2673  df-id 2842  df-xp 3191  df-rel 3192

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