Theorem weinxp 5186

Description: Intersection of well-ordering with Cartesian product of its field. (Contributed by NM, 9-Mar-1997.) (Revised by Mario Carneiro, 10-Jul-2014.)

Assertion

Ref	Expression
weinxp	|- ( R We A <-> ( R i^i ( A X. A ) ) We A )

Proof of Theorem weinxp

Step	Hyp	Ref	Expression
1		frinxp 5184	. . 3 |- ( R Fr A <-> ( R i^i ( A X. A ) ) Fr A )
2		soinxp 5183	. . 3 |- ( R Or A <-> ( R i^i ( A X. A ) ) Or A )
3	1, 2	anbi12i 733	. 2 |- ( ( R Fr A /\ R Or A ) <-> ( ( R i^i ( A X. A ) ) Fr A /\ ( R i^i ( A X. A ) ) Or A ) )
4		df-we 5075	. 2 |- ( R We A <-> ( R Fr A /\ R Or A ) )
5		df-we 5075	. 2 |- ( ( R i^i ( A X. A ) ) We A <-> ( ( R i^i ( A X. A ) ) Fr A /\ ( R i^i ( A X. A ) ) Or A ) )
6	3, 4, 5	3bitr4i 292	1 |- ( R We A <-> ( R i^i ( A X. A ) ) We A )

Syntax hints:   <-> wb 196   /\ wa 384   i^i cin 3573   Or wor 5034   Fr wfr 5070   We wwe 5072   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120

This theorem is referenced by:  wemapwe  8594  infxpenlem  8836  dfac8b  8854  ac10ct  8857  canthwelem  9472  ltbwe  19472  vitali  23382  fin2so  33396  dnwech  37618  aomclem5  37628