Theorem supxrunb2 11613

Description: The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)

Assertion

Ref	Expression
supxrunb2	|- ( A C_ RR* -> ( A. x e. RR E. y e. A x < y <-> sup ( A , RR* , < ) = +oo ) )

Distinct variable group:   x, y, A

Proof of Theorem supxrunb2

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3428	. . . . . . . 8 |- ( A C_ RR* -> ( z e. A -> z e. RR* ) )
2		pnfnlt 11437	. . . . . . . 8 |- ( z e. RR* -> -. +oo < z )
3	1, 2	syl6 34	. . . . . . 7 |- ( A C_ RR* -> ( z e. A -> -. +oo < z ) )
4	3	ralrimiv 2802	. . . . . 6 |- ( A C_ RR* -> A. z e. A -. +oo < z )
5	4	adantr 467	. . . . 5 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A x < y ) -> A. z e. A -. +oo < z )
6		breq1 4408	. . . . . . . . . . . . . . 15 |- ( x = z -> ( x < y <-> z < y ) )
7	6	rexbidv 2903	. . . . . . . . . . . . . 14 |- ( x = z -> ( E. y e. A x < y <-> E. y e. A z < y ) )
8	7	rspcva 3150	. . . . . . . . . . . . 13 |- ( ( z e. RR /\ A. x e. RR E. y e. A x < y ) -> E. y e. A z < y )
9	8	adantrr 724	. . . . . . . . . . . 12 |- ( ( z e. RR /\ ( A. x e. RR E. y e. A x < y /\ A C_ RR* ) ) -> E. y e. A z < y )
10	9	ancoms 455	. . . . . . . . . . 11 |- ( ( ( A. x e. RR E. y e. A x < y /\ A C_ RR* ) /\ z e. RR ) -> E. y e. A z < y )
11	10	exp31 609	. . . . . . . . . 10 |- ( A. x e. RR E. y e. A x < y -> ( A C_ RR* -> ( z e. RR -> E. y e. A z < y ) ) )
12	11	a1dd 47	. . . . . . . . 9 |- ( A. x e. RR E. y e. A x < y -> ( A C_ RR* -> ( z < +oo -> ( z e. RR -> E. y e. A z < y ) ) ) )
13	12	com4r 89	. . . . . . . 8 |- ( z e. RR -> ( A. x e. RR E. y e. A x < y -> ( A C_ RR* -> ( z < +oo -> E. y e. A z < y ) ) ) )
14	13	com13 83	. . . . . . 7 |- ( A C_ RR* -> ( A. x e. RR E. y e. A x < y -> ( z e. RR -> ( z < +oo -> E. y e. A z < y ) ) ) )
15	14	imp 431	. . . . . 6 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A x < y ) -> ( z e. RR -> ( z < +oo -> E. y e. A z < y ) ) )
16	15	ralrimiv 2802	. . . . 5 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A x < y ) -> A. z e. RR ( z < +oo -> E. y e. A z < y ) )
17	5, 16	jca 535	. . . 4 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A x < y ) -> ( A. z e. A -. +oo < z /\ A. z e. RR ( z < +oo -> E. y e. A z < y ) ) )
18		pnfxr 11419	. . . . 5 |- +oo e. RR*
19		supxr 11605	. . . . 5 |- ( ( ( A C_ RR* /\ +oo e. RR* ) /\ ( A. z e. A -. +oo < z /\ A. z e. RR ( z < +oo -> E. y e. A z < y ) ) ) -> sup ( A , RR* , < ) = +oo )
20	18, 19	mpanl2 688	. . . 4 |- ( ( A C_ RR* /\ ( A. z e. A -. +oo < z /\ A. z e. RR ( z < +oo -> E. y e. A z < y ) ) ) -> sup ( A , RR* , < ) = +oo )
21	17, 20	syldan 473	. . 3 |- ( ( A C_ RR* /\ A. x e. RR E. y e. A x < y ) -> sup ( A , RR* , < ) = +oo )
22	21	ex 436	. 2 |- ( A C_ RR* -> ( A. x e. RR E. y e. A x < y -> sup ( A , RR* , < ) = +oo ) )
23		rexr 9691	. . . . . . 7 |- ( x e. RR -> x e. RR* )
24	23	ad2antlr 734	. . . . . 6 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> x e. RR* )
25		ltpnf 11429	. . . . . . . . 9 |- ( x e. RR -> x < +oo )
26		breq2 4409	. . . . . . . . 9 |- ( sup ( A , RR* , < ) = +oo -> ( x < sup ( A , RR* , < ) <-> x < +oo ) )
27	25, 26	syl5ibr 225	. . . . . . . 8 |- ( sup ( A , RR* , < ) = +oo -> ( x e. RR -> x < sup ( A , RR* , < ) ) )
28	27	impcom 432	. . . . . . 7 |- ( ( x e. RR /\ sup ( A , RR* , < ) = +oo ) -> x < sup ( A , RR* , < ) )
29	28	adantll 721	. . . . . 6 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> x < sup ( A , RR* , < ) )
30		xrltso 11447	. . . . . . . 8 |- < Or RR*
31	30	a1i 11	. . . . . . 7 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> < Or RR* )
32		xrsupss 11601	. . . . . . . 8 |- ( A C_ RR* -> E. z e. RR* ( A. w e. A -. z < w /\ A. w e. RR* ( w < z -> E. y e. A w < y ) ) )
33	32	ad2antrr 733	. . . . . . 7 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> E. z e. RR* ( A. w e. A -. z < w /\ A. w e. RR* ( w < z -> E. y e. A w < y ) ) )
34	31, 33	suplub 7979	. . . . . 6 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> ( ( x e. RR* /\ x < sup ( A , RR* , < ) ) -> E. y e. A x < y ) )
35	24, 29, 34	mp2and 686	. . . . 5 |- ( ( ( A C_ RR* /\ x e. RR ) /\ sup ( A , RR* , < ) = +oo ) -> E. y e. A x < y )
36	35	exp31 609	. . . 4 |- ( A C_ RR* -> ( x e. RR -> ( sup ( A , RR* , < ) = +oo -> E. y e. A x < y ) ) )
37	36	com23 81	. . 3 |- ( A C_ RR* -> ( sup ( A , RR* , < ) = +oo -> ( x e. RR -> E. y e. A x < y ) ) )
38	37	ralrimdv 2806	. 2 |- ( A C_ RR* -> ( sup ( A , RR* , < ) = +oo -> A. x e. RR E. y e. A x < y ) )
39	22, 38	impbid 194	1 |- ( A C_ RR* -> ( A. x e. RR E. y e. A x < y <-> sup ( A , RR* , < ) = +oo ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1446   e. wcel 1889   A.wral 2739   E.wrex 2740   C_ wss 3406   class class class wbr 4405   Or wor 4757   supcsup 7959   RRcr 9543   +oocpnf 9677   RR*cxr 9679   < clt 9680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7961  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868

This theorem is referenced by:  supxrbnd2  11615  supxrbnd  11621  suplesup  37572  sge0pnffigt  38248