Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  frinxp Structured version   Unicode version

Theorem frinxp 5074
 Description: Intersection of well-founded relation with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)
Assertion
Ref Expression
frinxp

Proof of Theorem frinxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3493 . . . . . . . . . . 11
2 ssel 3493 . . . . . . . . . . 11
31, 2anim12d 563 . . . . . . . . . 10
4 brinxp 5071 . . . . . . . . . . 11
54ancoms 453 . . . . . . . . . 10
63, 5syl6 33 . . . . . . . . 9
76impl 620 . . . . . . . 8
87notbid 294 . . . . . . 7
98ralbidva 2893 . . . . . 6
109rexbidva 2965 . . . . 5
1110adantr 465 . . . 4
1211pm5.74i 245 . . 3
1312albii 1641 . 2
14 df-fr 4847 . 2
15 df-fr 4847 . 2
1613, 14, 153bitr4i 277 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 184   wa 369  wal 1393   wcel 1819   wne 2652  wral 2807  wrex 2808   cin 3470   wss 3471  c0 3793   class class class wbr 4456   wfr 4844   cxp 5006 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-fr 4847  df-xp 5014 This theorem is referenced by:  weinxp  5076
 Copyright terms: Public domain W3C validator