Theorem elixx3g 9837

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 399	. 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 970	. . 3 |- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) <-> ( ( A e. RR* /\ B e. RR* ) /\ C e. RR* ) )
3	2	anbi1i 454	. 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 6036	. . 3 |- ( C e. ( A O B ) -> ( A e. RR* /\ B e. RR* ) )
6	4	elixx1 9833	. . . 4 |- ( ( A e. RR* /\ B e. RR* ) -> ( C e. ( A O B ) <-> ( C e. RR* /\ A R C /\ C S B ) ) )
7		3anass 972	. . . 4 |- ( ( C e. RR* /\ A R C /\ C S B ) <-> ( C e. RR* /\ ( A R C /\ C S B ) ) )
8	6, 7	bitrdi 195	. . 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 452	. 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 212	1 |- ( C e. ( A O B ) <-> ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ ( A R C /\ C S B ) ) )

Syntax hints:   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   {crab 2448   class class class wbr 3982  (class class class)co 5842   e. cmpo 5844   RR*cxr 7932

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937

This theorem is referenced by:  ixxss1  9840  ixxss2  9841  ixxss12  9842  elioo3g  9846  iccss2  9880  iccssico2  9883  elicore  10202