Theorem leweon 8834

Description: Lexicographical order is a well-ordering of On X. On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 8835, this order is not set-like, as the preimage of <. 1o , (/) >. is the proper class ( { (/) } X. On ). (Contributed by Mario Carneiro, 9-Mar-2013.)

Hypothesis

Ref	Expression
leweon.1	|- L = { <. x , y >. | ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) e. ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) e. ( 2nd ` y ) ) ) ) }

Assertion

Ref	Expression
leweon	|- L We ( On X. On )

Distinct variable group:   x, y

Allowed substitution hints:   L( x, y)

Proof of Theorem leweon

Step	Hyp	Ref	Expression
1		epweon 6983	. 2 |- _E We On
2		leweon.1	. . . 4 |- L = { <. x , y >. | ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) e. ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) e. ( 2nd ` y ) ) ) ) }
3		fvex 6201	. . . . . . . 8 |- ( 1st ` y ) e. _V
4	3	epelc 5031	. . . . . . 7 |- ( ( 1st ` x ) _E ( 1st ` y ) <-> ( 1st ` x ) e. ( 1st ` y ) )
5		fvex 6201	. . . . . . . . 9 |- ( 2nd ` y ) e. _V
6	5	epelc 5031	. . . . . . . 8 |- ( ( 2nd ` x ) _E ( 2nd ` y ) <-> ( 2nd ` x ) e. ( 2nd ` y ) )
7	6	anbi2i 730	. . . . . . 7 |- ( ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) _E ( 2nd ` y ) ) <-> ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) e. ( 2nd ` y ) ) )
8	4, 7	orbi12i 543	. . . . . 6 |- ( ( ( 1st ` x ) _E ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) _E ( 2nd ` y ) ) ) <-> ( ( 1st ` x ) e. ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) e. ( 2nd ` y ) ) ) )
9	8	anbi2i 730	. . . . 5 |- ( ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) _E ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) _E ( 2nd ` y ) ) ) ) <-> ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) e. ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) e. ( 2nd ` y ) ) ) ) )
10	9	opabbii 4717	. . . 4 |- { <. x , y >. | ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) _E ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) _E ( 2nd ` y ) ) ) ) } = { <. x , y >. | ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) e. ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) e. ( 2nd ` y ) ) ) ) }
11	2, 10	eqtr4i 2647	. . 3 |- L = { <. x , y >. | ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) _E ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) _E ( 2nd ` y ) ) ) ) }
12	11	wexp 7291	. 2 |- ( ( _E We On /\ _E We On ) -> L We ( On X. On ) )
13	1, 1, 12	mp2an 708	1 |- L We ( On X. On )

Syntax hints:   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   {copab 4712   _E cep 5028   We wwe 5072   X. cxp 5112   Oncon0 5723   ` 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-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  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-ord 5726  df-on 5727  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  r0weon  8835