Theorem wexp 7291

Description: A lexicographical ordering of two well-ordered classes. (Contributed by Scott Fenton, 17-Mar-2011.) (Revised by Mario Carneiro, 7-Mar-2013.)

Hypothesis

Ref	Expression
wexp.1	|- T = { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) /\ ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) }

Assertion

Ref	Expression
wexp	|- ( ( R We A /\ S We B ) -> T We ( A X. B ) )

Distinct variable groups:   x, A, y   x, B, y   x, R, y   x, S, y

Allowed substitution hints:   T( x, y)

Proof of Theorem wexp

Step	Hyp	Ref	Expression
1		wefr 5104	. . 3 |- ( R We A -> R Fr A )
2		wefr 5104	. . 3 |- ( S We B -> S Fr B )
3		wexp.1	. . . 4 |- T = { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) /\ ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) }
4	3	frxp 7287	. . 3 |- ( ( R Fr A /\ S Fr B ) -> T Fr ( A X. B ) )
5	1, 2, 4	syl2an 494	. 2 |- ( ( R We A /\ S We B ) -> T Fr ( A X. B ) )
6		weso 5105	. . 3 |- ( R We A -> R Or A )
7		weso 5105	. . 3 |- ( S We B -> S Or B )
8	3	soxp 7290	. . 3 |- ( ( R Or A /\ S Or B ) -> T Or ( A X. B ) )
9	6, 7, 8	syl2an 494	. 2 |- ( ( R We A /\ S We B ) -> T Or ( A X. B ) )
10		df-we 5075	. 2 |- ( T We ( A X. B ) <-> ( T Fr ( A X. B ) /\ T Or ( A X. B ) ) )
11	5, 9, 10	sylanbrc 698	1 |- ( ( R We A /\ S We B ) -> T We ( A X. B ) )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   {copab 4712   Or wor 5034   Fr wfr 5070   We wwe 5072   X. cxp 5112   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  fnwelem  7292  leweon  8834