Theorem xrssre 39564

Description: A subset of extended reals that does not contain +oo and -oo is a subset of the reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypotheses

Ref	Expression
xrssre.1	|- ( ph -> A C_ RR* )
xrssre.2	|- ( ph -> -. +oo e. A )
xrssre.3	|- ( ph -> -. -oo e. A )

Assertion

Ref	Expression
xrssre	|- ( ph -> A C_ RR )

Proof of Theorem xrssre

Step	Hyp	Ref	Expression
1		xrssre.1	. . . . 5 |- ( ph -> A C_ RR* )
2		ssxr 10107	. . . . 5 |- ( A C_ RR* -> ( A C_ RR \/ +oo e. A \/ -oo e. A ) )
3	1, 2	syl 17	. . . 4 |- ( ph -> ( A C_ RR \/ +oo e. A \/ -oo e. A ) )
4		3orass 1040	. . . 4 |- ( ( A C_ RR \/ +oo e. A \/ -oo e. A ) <-> ( A C_ RR \/ ( +oo e. A \/ -oo e. A ) ) )
5	3, 4	sylib 208	. . 3 |- ( ph -> ( A C_ RR \/ ( +oo e. A \/ -oo e. A ) ) )
6	5	orcomd 403	. 2 |- ( ph -> ( ( +oo e. A \/ -oo e. A ) \/ A C_ RR ) )
7		xrssre.2	. . . 4 |- ( ph -> -. +oo e. A )
8		xrssre.3	. . . 4 |- ( ph -> -. -oo e. A )
9	7, 8	jca 554	. . 3 |- ( ph -> ( -. +oo e. A /\ -. -oo e. A ) )
10		ioran 511	. . 3 |- ( -. ( +oo e. A \/ -oo e. A ) <-> ( -. +oo e. A /\ -. -oo e. A ) )
11	9, 10	sylibr 224	. 2 |- ( ph -> -. ( +oo e. A \/ -oo e. A ) )
12		df-or 385	. . 3 |- ( ( ( +oo e. A \/ -oo e. A ) \/ A C_ RR ) <-> ( -. ( +oo e. A \/ -oo e. A ) -> A C_ RR ) )
13	12	biimpi 206	. 2 |- ( ( ( +oo e. A \/ -oo e. A ) \/ A C_ RR ) -> ( -. ( +oo e. A \/ -oo e. A ) -> A C_ RR ) )
14	6, 11, 13	sylc 65	1 |- ( ph -> A C_ RR )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   \/ w3o 1036   e. wcel 1990   C_ wss 3574   RRcr 9935   +oocpnf 10071   -oocmnf 10072   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-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  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-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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078

This theorem is referenced by:  supminfxr2  39699