Theorem elixx3g 9993

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 6122	. . 3 |- ( C e. ( A O B ) -> ( A e. RR* /\ B e. RR* ) )
6	4	elixx1 9989	. . . 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 2167   {crab 2479   class class class wbr 4034  (class class class)co 5925   e. cmpo 5927   RR*cxr 8077

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082

This theorem is referenced by:  ixxss1  9996  ixxss2  9997  ixxss12  9998  elioo3g  10002  iccss2  10036  iccssico2  10039  elicore  10373