Theorem suplub 7940

Description: A supremum is the least upper bound. See also supcl 7938 and supub 7939. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.)

Hypotheses

Ref	Expression
supmo.1
supcl.2

Assertion

Ref	Expression
suplub

Distinct variable groups:   ,,,   ,,,   ,,,   ,

Proof of Theorem suplub

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 461	. . . . . . 7
2		breq1 4455	. . . . . . . . 9
3		breq1 4455	. . . . . . . . . 10
4	3	rexbidv 2968	. . . . . . . . 9
5	2, 4	imbi12d 320	. . . . . . . 8
6	5	cbvralv 3084	. . . . . . 7
7	1, 6	sylib 196	. . . . . 6
8	7	a1i 11	. . . . 5
9	8	ss2rabi 3581	. . . 4
10		supmo.1	. . . . . 6
11	10	supval2 7935	. . . . 5
12		supcl.2	. . . . . . 7
13	10, 12	supeu 7934	. . . . . 6
14		riotacl2 6271	. . . . . 6
15	13, 14	syl 16	. . . . 5
16	11, 15	eqeltrd 2545	. . . 4
17	9, 16	sseldi 3501	. . 3
18		breq2 4456	. . . . . . 7
19	18	imbi1d 317	. . . . . 6
20	19	ralbidv 2896	. . . . 5
21	20	elrab 3257	. . . 4
22	21	simprbi 464	. . 3
23	17, 22	syl 16	. 2
24		breq1 4455	. . . . 5
25		breq1 4455	. . . . . 6
26	25	rexbidv 2968	. . . . 5
27	24, 26	imbi12d 320	. . . 4
28	27	rspccv 3207	. . 3
29	28	impd 431	. 2
30	23, 29	syl 16	1

Syntax hints:  -.wn 3  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  E.wrex 2808  E!wreu 2809  {crab 2811   class class class wbr 4452  Orwor 4804  iota_crio 6256  supcsup 7920

This theorem is referenced by:  suplub2  7941  supnub  7942  supiso  7954  supxrun  11536  supxrunb1  11540  supxrunb2  11541  gtinf  30137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-po 4805  df-so 4806  df-iota 5556  df-riota 6257  df-sup 7921