Theorem upgr1elem1 15977

Description: Lemma for upgr1edc 15978. (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 4091	. . . 4 |- ( x = { B , C } -> ( x ~~ 1o <-> { B , C } ~~ 1o ) )
2		breq1 4091	. . . 4 |- ( x = { B , C } -> ( x ~~ 2o <-> { B , C } ~~ 2o ) )
3	1, 2	orbi12d 800	. . 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 7399	. . . 4 |- ( ( B e. W /\ C e. X /\ DECID B = C ) -> ( { B , C } ~~ 1o \/ { B , C } ~~ 2o ) )
9	5, 6, 7, 8	syl3anc 1273	. . 3 |- ( ph -> ( { B , C } ~~ 1o \/ { B , C } ~~ 2o ) )
10	3, 4, 9	elrabd 2964	. 2 |- ( ph -> { B , C } e. { x e. S | ( x ~~ 1o \/ x ~~ 2o ) } )
11	10	snssd 3818	1 |- ( ph -> { { B , C } } C_ { x e. S | ( x ~~ 1o \/ x ~~ 2o ) } )

Syntax hints:   -> wi 4   \/ wo 715  DECID wdc 841   = wceq 1397   e. wcel 2202   {crab 2514   C_ wss 3200   {csn 3669   {cpr 3670   class class class wbr 4088   1oc1o 6575   2oc2o 6576   ~~ cen 6907

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6582  df-2o 6583  df-er 6702  df-en 6910

This theorem is referenced by:  upgr1edc  15978  uspgr1edc  16097