Theorem supub 7939

Description: A supremum is an upper bound. See also supcl 7938 and suplub 7940.

This proof demonstrates how to expand an iota-based definition (df-iota 5556) using riotacl2 6271.

(Contributed by NM, 12-Oct-2004.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Hypotheses

Ref	Expression
supmo.1
supcl.2

Assertion

Ref	Expression
supub

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

Proof of Theorem supub

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 457	. . . . . 6
2	1	a1i 11	. . . . 5
3	2	ss2rabi 3581	. . . 4
4		supmo.1	. . . . . 6
5	4	supval2 7935	. . . . 5
6		supcl.2	. . . . . . 7
7	4, 6	supeu 7934	. . . . . 6
8		riotacl2 6271	. . . . . 6
9	7, 8	syl 16	. . . . 5
10	5, 9	eqeltrd 2545	. . . 4
11	3, 10	sseldi 3501	. . 3
12		breq2 4456	. . . . . . . 8
13	12	notbid 294	. . . . . . 7
14	13	cbvralv 3084	. . . . . 6
15		breq1 4455	. . . . . . . 8
16	15	notbid 294	. . . . . . 7
17	16	ralbidv 2896	. . . . . 6
18	14, 17	syl5bb 257	. . . . 5
19	18	elrab 3257	. . . 4
20	19	simprbi 464	. . 3
21	11, 20	syl 16	. 2
22		breq2 4456	. . . 4
23	22	notbid 294	. . 3
24	23	rspccv 3207	. 2
25	21, 24	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  supmaxOLD  7946  supgtoreq  7949  supiso  7954  suprub  10529  infmrlb  10549  suprzub  11202  supxrun  11536  supxrub  11545  infmxrlb  11554  dgrub  22631  supssd  27527  ssnnssfz  27597  oddpwdc  28293  ballotlemimin  28444  ballotlemfrcn0  28468  wsuclb  29384  itg2addnclem  30066  supubt  30230  ssnn0ssfz  32938

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