Theorem wlogle 10561

Description: If the predicate ch ( x , y ) is symmetric under interchange of x , y, then "without loss of generality" we can assume that x <_ y. (Contributed by Mario Carneiro, 18-Aug-2014.) (Revised by Mario Carneiro, 11-Sep-2014.)

Hypotheses

Ref	Expression
wlogle.1	|- ( ( z = x /\ w = y ) -> ( ps <-> ch ) )
wlogle.2	|- ( ( z = y /\ w = x ) -> ( ps <-> th ) )
wlogle.3	|- ( ph -> S C_ RR )
wlogle.4	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( ch <-> th ) )
wlogle.5	|- ( ( ph /\ ( x e. S /\ y e. S /\ x <_ y ) ) -> ch )

Assertion

Ref	Expression
wlogle	|- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ch )

Distinct variable groups:   x, w, y, z, ph   w, S, x, y, z   ps, x, y   ch, w, z

Allowed substitution hints:   ps( z, w)   ch( x, y)   th( x, y, z, w)

Proof of Theorem wlogle

Step	Hyp	Ref	Expression
1		wlogle.1	. 2 |- ( ( z = x /\ w = y ) -> ( ps <-> ch ) )
2		wlogle.2	. 2 |- ( ( z = y /\ w = x ) -> ( ps <-> th ) )
3		wlogle.3	. 2 |- ( ph -> S C_ RR )
4		wlogle.5	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ x <_ y ) ) -> ch )
5		wlogle.4	. . . 4 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ( ch <-> th ) )
6	5	3adantr3 1222	. . 3 |- ( ( ph /\ ( x e. S /\ y e. S /\ x <_ y ) ) -> ( ch <-> th ) )
7	4, 6	mpbid 222	. 2 |- ( ( ph /\ ( x e. S /\ y e. S /\ x <_ y ) ) -> th )
8	1, 2, 3, 7, 4	wloglei 10560	1 |- ( ( ph /\ ( x e. S /\ y e. S ) ) -> ch )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   e. wcel 1990   C_ wss 3574   class class class wbr 4653   RRcr 9935   <_ cle 10075

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  ax-resscn 9993  ax-pre-lttri 10010

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080

This theorem is referenced by:  vdwlem12  15696  iundisj2  23317  volcn  23374  dvlip  23756  ftc1a  23800  iundisj2f  29403  iundisj2fi  29556  erdszelem9  31181  ftc1anc  33493