Theorem elixx1 9854

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 9853	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A O B ) = { z e. RR* | ( A R z /\ z S B ) } )
3	2	eleq2d 2240	. 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 3993	. . . . 5 |- ( z = C -> ( A R z <-> A R C ) )
5		breq1 3992	. . . . 5 |- ( z = C -> ( z S B <-> C S B ) )
6	4, 5	anbi12d 470	. . . 4 |- ( z = C -> ( ( A R z /\ z S B ) <-> ( A R C /\ C S B ) ) )
7	6	elrab 2886	. . 3 |- ( C e. { z e. RR* | ( A R z /\ z S B ) } <-> ( C e. RR* /\ ( A R C /\ C S B ) ) )
8		3anass 977	. . 3 |- ( ( C e. RR* /\ A R C /\ C S B ) <-> ( C e. RR* /\ ( A R C /\ C S B ) ) )
9	7, 8	bitr4i 186	. 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 195	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 103   <-> wb 104   /\ w3a 973   = wceq 1348   e. wcel 2141   {crab 2452   class class class wbr 3989  (class class class)co 5853   e. cmpo 5855   RR*cxr 7953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958

This theorem is referenced by:  elixx3g  9858  ixxssixx  9859  ixxdisj  9860  ixxss1  9861  ixxss2  9862  ixxss12  9863  elioo1  9868  elioc1  9879  elico1  9880  elicc1  9881