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Theorem suppr 7235
 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 variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 955 . 2
2 ifcl 3614 . . 3
4 breq1 4042 . . . . . 6
54notbid 285 . . . . 5
6 breq1 4042 . . . . . 6
76notbid 285 . . . . 5
8 sonr 4351 . . . . . . 7
983adant3 975 . . . . . 6
109adantr 451 . . . . 5
11 simpr 447 . . . . 5
125, 7, 10, 11ifbothda 3608 . . . 4
13 breq1 4042 . . . . . 6
1413notbid 285 . . . . 5
15 breq1 4042 . . . . . 6
1615notbid 285 . . . . 5
17 so2nr 4354 . . . . . . . . 9
18173impb 1147 . . . . . . . 8
19183com23 1157 . . . . . . 7
20 imnan 411 . . . . . . 7
2119, 20sylibr 203 . . . . . 6
2221imp 418 . . . . 5
23 sonr 4351 . . . . . . 7
24233adant2 974 . . . . . 6
2524adantr 451 . . . . 5
2614, 16, 22, 25ifbothda 3608 . . . 4
27 breq2 4043 . . . . . . 7
2827notbid 285 . . . . . 6
29 breq2 4043 . . . . . . 7
3029notbid 285 . . . . . 6
3128, 30ralprg 3695 . . . . 5
32313adant1 973 . . . 4
3312, 26, 32mpbir2and 888 . . 3
3433r19.21bi 2654 . 2
35 ifpr 3694 . . . . . 6
36353adant1 973 . . . . 5
3736adantr 451 . . . 4
38 breq2 4043 . . . . 5
3938rspcev 2897 . . . 4
4037, 39sylan 457 . . 3
4140anasss 628 . 2
421, 3, 34, 41eqsupd 7224 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wa 358   w3a 934   wceq 1632   wcel 1696  wral 2556  wrex 2557  cif 3578  cpr 3654   class class class wbr 4039   wor 4329  csup 7209 This theorem is referenced by:  supsn  7236  tmsxpsval2  18101  esumsn  23452 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-po 4330  df-so 4331  df-iota 5235  df-riota 6320  df-sup 7210
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