Theorem brrelex12 4324

Description: A true binary relation on a relation implies the arguments are sets. (This is a property of our ordered pair definition.) (Contributed by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
brrelex12	⊢ ((Rel 𝑅 ∧ A𝑅B) → (A ∈ V ∧ B ∈ V))

Proof of Theorem brrelex12

Step	Hyp	Ref	Expression
1		df-rel 4295	. . . . 5 ⊢ (Rel 𝑅 ↔ 𝑅 ⊆ (V × V))
2	1	biimpi 113	. . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (V × V))
3	2	ssbrd 3796	. . 3 ⊢ (Rel 𝑅 → (A𝑅B → A(V × V)B))
4	3	imp 115	. 2 ⊢ ((Rel 𝑅 ∧ A𝑅B) → A(V × V)B)
5		brxp 4318	. 2 ⊢ (A(V × V)B ↔ (A ∈ V ∧ B ∈ V))
6	4, 5	sylib 127	1 ⊢ ((Rel 𝑅 ∧ A𝑅B) → (A ∈ V ∧ B ∈ V))

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911   class class class wbr 3755   × cxp 4286  Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295

This theorem is referenced by:  brrelex  4325  brrelex2  4326  relbrcnvg  4647  ovprc  5482  ersym  6054  relelec  6082  encv  6163