Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inxpex Structured version   Visualization version   GIF version

Theorem inxpex 35128
 Description: Sufficient condition for an intersection with a Cartesian product to be a set. (Contributed by Peter Mazsa, 10-May-2019.)
Assertion
Ref Expression
inxpex ((𝑅𝑊 ∨ (𝐴𝑈𝐵𝑉)) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)

Proof of Theorem inxpex
StepHypRef Expression
1 inex1g 5114 . 2 (𝑅𝑊 → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
2 xpexg 7330 . . 3 ((𝐴𝑈𝐵𝑉) → (𝐴 × 𝐵) ∈ V)
3 inex2g 5115 . . 3 ((𝐴 × 𝐵) ∈ V → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
42, 3syl 17 . 2 ((𝐴𝑈𝐵𝑉) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
51, 4jaoi 852 1 ((𝑅𝑊 ∨ (𝐴𝑈𝐵𝑉)) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 842   ∈ wcel 2081  Vcvv 3437   ∩ cin 3858   × cxp 5441 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-opab 5025  df-xp 5449  df-rel 5450 This theorem is referenced by:  xrninxpex  35173
 Copyright terms: Public domain W3C validator