Theorem elixx1 12184

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 12183	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( A O B ) = { z e. RR* | ( A R z /\ z S B ) } )
3	2	eleq2d 2687	. 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 4657	. . . . 5 |- ( z = C -> ( A R z <-> A R C ) )
5		breq1 4656	. . . . 5 |- ( z = C -> ( z S B <-> C S B ) )
6	4, 5	anbi12d 747	. . . 4 |- ( z = C -> ( ( A R z /\ z S B ) <-> ( A R C /\ C S B ) ) )
7	6	elrab 3363	. . 3 |- ( C e. { z e. RR* | ( A R z /\ z S B ) } <-> ( C e. RR* /\ ( A R C /\ C S B ) ) )
8		3anass 1042	. . 3 |- ( ( C e. RR* /\ A R C /\ C S B ) <-> ( C e. RR* /\ ( A R C /\ C S B ) ) )
9	7, 8	bitr4i 267	. 2 |- ( C e. { z e. RR* | ( A R z /\ z S B ) } <-> ( C e. RR* /\ A R C /\ C S B ) )
10	3, 9	syl6bb 276	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   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   class class class wbr 4653  (class class class)co 6650   |-> cmpt2 6652   RR*cxr 10073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-xr 10078

This theorem is referenced by:  elixx3g  12188  ixxssixx  12189  ixxdisj  12190  ixxun  12191  ixxss1  12193  ixxss2  12194  ixxss12  12195  ixxub  12196  ixxlb  12197  elioo1  12215  elioc1  12217  elico1  12218  elicc1  12219