Theorem brrelex 2446

Description: A true binary relation on a relation implies the first argument is a set. (This is a property of our ordered pair definition.)

Assertion

Ref	Expression
brrelex	⊢ ((Rel R ∧ ARB) → A ∈ V)

Proof of Theorem brrelex

Step	Hyp	Ref	Expression
1		df-rel 2425	. . 3 ⊢ (Rel R ↔ R ⊆ (V × V))
2		ssel 1502	. . . . 5 ⊢ (R ⊆ (V × V) → (⟨A, B⟩ ∈ R → ⟨A, B⟩ ∈ (V × V)))
3		df-br 2063	. . . . 5 ⊢ (ARB ↔ ⟨A, B⟩ ∈ R)
4	2, 3	syl5ib 181	. . . 4 ⊢ (R ⊆ (V × V) → (ARB → ⟨A, B⟩ ∈ (V × V)))
5		opelxpex 2445	. . . 4 ⊢ (⟨A, B⟩ ∈ (V × V) → A ∈ V)
6	4, 5	syl6 23	. . 3 ⊢ (R ⊆ (V × V) → (ARB → A ∈ V))
7	1, 6	sylbi 174	. 2 ⊢ (Rel R → (ARB → A ∈ V))
8	7	imp 277	1 ⊢ ((Rel R ∧ ARB) → A ∈ V)

Syntax hints:   → wi 2   ∧ wa 196   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ⟨cop 1810   class class class wbr 2054   × cxp 2408  Rel wrel 2415

This theorem is referenced by:  brrelexi 2447  funbrfv 2852

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425

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