Theorem elixx3g 9933

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
ixxssxr.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, R   x, S, y, z   x, A, y, z   x, B, y, z   x, C, y, z

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

Proof of Theorem elixx3g

Step	Hyp	Ref	Expression
1		anass 401	. 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 982	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) <-> ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) )
3	2	anbi1i 458	. 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		ixxssxr.1	. . . 4 |- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
5	4	elmpocl 6092	. . 3 |- ( C e. ( A O B ) -> ( A e. RR* /\ B e. RR* ) )
6	4	elixx1 9929	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> ( C e. RR* /\ A R C /\ C S B ) ) )
7		3anass 984	. . . 4 |- ( ( C e. RR* /\ A R C /\ C S B ) <-> ( C e. RR* /\ ( A R C /\ C S B ) ) )
8	6, 7	bitrdi 196	. . 3 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> ( C e. RR* /\ ( A R C /\ C S B ) ) ) )
9	5, 8	biadan2 456	. 2 |- ( C e. ( A O B ) <-> ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ ( A R C /\ C S B ) ) ) )
10	1, 3, 9	3bitr4ri 213	1 |- ( C e. ( A O B ) <-> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A R C /\ C S B ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   {crab 2472   class class class wbr 4018  (class class class)co 5897   e. cmpo 5899   RR*cxr 8022

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-xr 8027

This theorem is referenced by:  ixxss1  9936  ixxss2  9937  ixxss12  9938  elioo3g  9942  iccss2  9976  iccssico2  9979  elicore  10299