Theorem elixx1 10093

Description: Membership in an interval of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.)

Hypothesis

Ref	Expression
ixx.1	|- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )

Assertion

Ref	Expression
elixx1	|- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> ( C e. RR* /\ A R C /\ C S B ) ) )

Distinct variable groups:   x, y, z, A   x, C, y, z   x, B, y, z   x, R, y, z   x, S, y, z

Allowed substitution hints:   O( x, y, z)

Proof of Theorem elixx1

Step	Hyp	Ref	Expression
1		ixx.1	. . . 4 |- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
2	1	ixxval 10092	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A O B ) = { z e. RR* | ( A R z /\ z S B ) } )
3	2	eleq2d 2299	. 2 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> C e. { z e. RR* | ( A R z /\ z S B ) } ) )
4		breq2 4087	. . . . 5 |- ( z = C -> ( A R z <-> A R C ) )
5		breq1 4086	. . . . 5 |- ( z = C -> ( z S B <-> C S B ) )
6	4, 5	anbi12d 473	. . . 4 |- ( z = C -> ( ( A R z /\ z S B ) <-> ( A R C /\ C S B ) ) )
7	6	elrab 2959	. . 3 |- ( C e. { z e. RR* | ( A R z /\ z S B ) } <-> ( C e. RR* /\ ( A R C /\ C S B ) ) )
8		3anass 1006	. . 3 |- ( ( C e. RR* /\ A R C /\ C S B ) <-> ( C e. RR* /\ ( A R C /\ C S B ) ) )
9	7, 8	bitr4i 187	. 2 |- ( C e. { z e. RR* | ( A R z /\ z S B ) } <-> ( C e. RR* /\ A R C /\ C S B ) )
10	3, 9	bitrdi 196	1 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> ( C e. RR* /\ A R C /\ C S B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   class class class wbr 4083  (class class class)co 6001   e. cmpo 6003   RR*cxr 8180

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185

This theorem is referenced by:  elixx3g  10097  ixxssixx  10098  ixxdisj  10099  ixxss1  10100  ixxss2  10101  ixxss12  10102  elioo1  10107  elioc1  10118  elico1  10119  elicc1  10120