Theorem elixx1 10249

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 10248	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A O B ) = { z e. RR* | ( A R z /\ z S B ) } )
3	2	eleq2d 2304	. 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 4118	. . . . 5 |- ( z = C -> ( A R z <-> A R C ) )
5		breq1 4117	. . . . 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 2976	. . 3 |- ( C e. { z e. RR* | ( A R z /\ z S B ) } <-> ( C e. RR* /\ ( A R C /\ C S B ) ) )
8		3anass 1009	. . 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 1005   = wceq 1398   e. wcel 2205   {crab 2526   class class class wbr 4114  (class class class)co 6058   e. cmpo 6060   RR*cxr 8323

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-iota 5317  df-fun 5359  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328

This theorem is referenced by:  elixx3g  10253  ixxssixx  10254  ixxdisj  10255  ixxss1  10256  ixxss2  10257  ixxss12  10258  elioo1  10263  elioc1  10274  elico1  10275  elicc1  10276