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Theorem suppr 7465
 Description: The supremum of a pair. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
suppr

Proof of Theorem suppr
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simp1 957 . 2
2 ifcl 3767 . . 3
323adant1 975 . 2
4 ifpr 3848 . . 3
543adant1 975 . 2
6 breq1 4207 . . . . . 6
76notbid 286 . . . . 5
8 breq1 4207 . . . . . 6
98notbid 286 . . . . 5
10 sonr 4516 . . . . . . 7
11103adant3 977 . . . . . 6
1211adantr 452 . . . . 5
13 simpr 448 . . . . 5
147, 9, 12, 13ifbothda 3761 . . . 4
15 breq1 4207 . . . . . 6
1615notbid 286 . . . . 5
17 breq1 4207 . . . . . 6
1817notbid 286 . . . . 5
19 so2nr 4519 . . . . . . . . 9
20193impb 1149 . . . . . . . 8
21203com23 1159 . . . . . . 7
22 imnan 412 . . . . . . 7
2321, 22sylibr 204 . . . . . 6
2423imp 419 . . . . 5
25 sonr 4516 . . . . . . 7
26253adant2 976 . . . . . 6
2726adantr 452 . . . . 5
2816, 18, 24, 27ifbothda 3761 . . . 4
29 breq2 4208 . . . . . . 7
3029notbid 286 . . . . . 6
31 breq2 4208 . . . . . . 7
3231notbid 286 . . . . . 6
3330, 32ralprg 3849 . . . . 5
34333adant1 975 . . . 4
3514, 28, 34mpbir2and 889 . . 3
3635r19.21bi 2796 . 2
371, 3, 5, 36supmax 7462 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2697  cif 3731  cpr 3807   class class class wbr 4204   wor 4494  csup 7437 This theorem is referenced by:  supsn  7466  tmsxpsval2  18561  esumsn  24448 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-po 4495  df-so 4496  df-iota 5410  df-riota 6541  df-sup 7438
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