Theorem elixx3g 12188

Description: Membership in a set of open intervals of extended reals. We use the fact that an operation's value is empty outside of its domain to show A e. RR* and B e. RR*. (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
elixx3g	|- ( C e. ( A O B ) <-> ( ( A e. RR* /\ B e. RR* /\ 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 elixx3g

Step	Hyp	Ref	Expression
1		anass 681	. 2 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ ( A R C /\ C S B ) ) <-> ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ ( A R C /\ C S B ) ) ) )
2		df-3an 1039	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) <-> ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) )
3	2	anbi1i 731	. 2 |- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A R C /\ C S B ) ) <-> ( ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) /\ ( A R C /\ C S B ) ) )
4		ixx.1	. . . . 5 |- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
5	4	elixx1 12184	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> ( C e. RR* /\ A R C /\ C S B ) ) )
6		3anass 1042	. . . . 5 |- ( ( C e. RR* /\ A R C /\ C S B ) <-> ( C e. RR* /\ ( A R C /\ C S B ) ) )
7		ibar 525	. . . . 5 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( C e. RR* /\ ( A R C /\ C S B ) ) <-> ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ ( A R C /\ C S B ) ) ) ) )
8	6, 7	syl5bb 272	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( ( C e. RR* /\ A R C /\ C S B ) <-> ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ ( A R C /\ C S B ) ) ) ) )
9	5, 8	bitrd 268	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ ( A R C /\ C S B ) ) ) ) )
10	4	ixxf 12185	. . . . . . 7 |- O : ( RR* X. RR* ) --> ~P RR*
11	10	fdmi 6052	. . . . . 6 |- dom O = ( RR* X. RR* )
12	11	ndmov 6818	. . . . 5 |- ( -. ( A e. RR* /\ B e. RR* ) -> ( A O B ) = (/) )
13	12	eleq2d 2687	. . . 4 |- ( -. ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> C e. (/) ) )
14		noel 3919	. . . . . 6 |- -. C e. (/)
15	14	pm2.21i 116	. . . . 5 |- ( C e. (/) -> ( A e. RR* /\ B e. RR* ) )
16		simpl 473	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ ( A R C /\ C S B ) ) ) -> ( A e. RR* /\ B e. RR* ) )
17	15, 16	pm5.21ni 367	. . . 4 |- ( -. ( A e. RR* /\ B e. RR* ) -> ( C e. (/) <-> ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ ( A R C /\ C S B ) ) ) ) )
18	13, 17	bitrd 268	. . 3 |- ( -. ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ ( A R C /\ C S B ) ) ) ) )
19	9, 18	pm2.61i 176	. 2 |- ( C e. ( A O B ) <-> ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ ( A R C /\ C S B ) ) ) )
20	1, 3, 19	3bitr4ri 293	1 |- ( C e. ( A O B ) <-> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A R C /\ C S B ) ) )

Syntax hints:   -. wn 3   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {crab 2916   (/)c0 3915   ~Pcpw 4158   class class class wbr 4653   X. cxp 5112  (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-pow 4843  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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-xr 10078

This theorem is referenced by:  ixxss1  12193  ixxss2  12194  ixxss12  12195  elioo3g  12204  elicore  12226  iccss2  12244  iccssico2  12247  xrtgioo  22609  ftc1anclem7  33491  ftc1anclem8  33492  ftc1anc  33493  eliocre  39734  lbioc  39739