Theorem upgr1elem1 16344

Description: Lemma for upgr1edc 16345. (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 4131	. . . 4 |- ( x = { B , C } -> ( x ~~ 1o <-> { B , C } ~~ 1o ) )
2		breq1 4131	. . . 4 |- ( x = { B , C } -> ( x ~~ 2o <-> { B , C } ~~ 2o ) )
3	1, 2	orbi12d 805	. . 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 7534	. . . 4 |- ( ( B e. W /\ C e. X /\ DECID B = C ) -> ( { B , C } ~~ 1o \/ { B , C } ~~ 2o ) )
9	5, 6, 7, 8	syl3anc 1278	. . 3 |- ( ph -> ( { B , C } ~~ 1o \/ { B , C } ~~ 2o ) )
10	3, 4, 9	elrabd 2984	. 2 |- ( ph -> { B , C } e. { x e. S | ( x ~~ 1o \/ x ~~ 2o ) } )
11	10	snssd 3858	1 |- ( ph -> { { B , C } } C_ { x e. S | ( x ~~ 1o \/ x ~~ 2o ) } )

Syntax hints:   -> wi 4   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   {crab 2532   C_ wss 3220   {csn 3708   {cpr 3709   class class class wbr 4128   1oc1o 6674   2oc2o 6675   ~~ cen 7014

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-1o 6681  df-2o 6682  df-er 6801  df-en 7017

This theorem is referenced by:  upgr1edc  16345  uspgr1edc  16464