Theorem upgr1elem1 16041

Description: Lemma for upgr1edc 16042. (Contributed by AV, 16-Oct-2020.) (Revised by Jim Kingdon, 6-Jan-2026.)

Hypotheses

Ref	Expression
upgr1elem.s	|- ( ph -> { B , C } e. S )
upgr1elem.b	|- ( ph -> B e. W )
upgr1elem.c	|- ( ph -> C e. X )
upgr1elem.dc	|- ( ph -> DECID B = C )

Assertion

Ref	Expression
upgr1elem1	|- ( ph -> { { B , C } } C_ { x e. S | ( x ~~ 1o \/ x ~~ 2o ) } )

Distinct variable groups:   x, B   x, C   x, S

Allowed substitution hints:   ph( x)   W( x)   X( x)

Proof of Theorem upgr1elem1

Step	Hyp	Ref	Expression
1		breq1 4096	. . . 4 |- ( x = { B , C } -> ( x ~~ 1o <-> { B , C } ~~ 1o ) )
2		breq1 4096	. . . 4 |- ( x = { B , C } -> ( x ~~ 2o <-> { B , C } ~~ 2o ) )
3	1, 2	orbi12d 801	. . 3 |- ( x = { B , C } -> ( ( x ~~ 1o \/ x ~~ 2o ) <-> ( { B , C } ~~ 1o \/ { B , C } ~~ 2o ) ) )
4		upgr1elem.s	. . 3 |- ( ph -> { B , C } e. S )
5		upgr1elem.b	. . . 4 |- ( ph -> B e. W )
6		upgr1elem.c	. . . 4 |- ( ph -> C e. X )
7		upgr1elem.dc	. . . 4 |- ( ph -> DECID B = C )
8		pr1or2 7442	. . . 4 |- ( ( B e. W /\ C e. X /\ DECID B = C ) -> ( { B , C } ~~ 1o \/ { B , C } ~~ 2o ) )
9	5, 6, 7, 8	syl3anc 1274	. . 3 |- ( ph -> ( { B , C } ~~ 1o \/ { B , C } ~~ 2o ) )
10	3, 4, 9	elrabd 2965	. 2 |- ( ph -> { B , C } e. { x e. S | ( x ~~ 1o \/ x ~~ 2o ) } )
11	10	snssd 3823	1 |- ( ph -> { { B , C } } C_ { x e. S | ( x ~~ 1o \/ x ~~ 2o ) } )

Syntax hints:   -> wi 4   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2202   {crab 2515   C_ wss 3201   {csn 3673   {cpr 3674   class class class wbr 4093   1oc1o 6618   2oc2o 6619   ~~ cen 6950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1o 6625  df-2o 6626  df-er 6745  df-en 6953

This theorem is referenced by:  upgr1edc  16042  uspgr1edc  16161