Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  suplub2 Structured version   Unicode version

Theorem suplub2 7466
 Description: Bidirectional form of suplub 7465. (Contributed by Mario Carneiro, 6-Sep-2014.)
Hypotheses
Ref Expression
supmo.1
supcl.2
suplub2.3
Assertion
Ref Expression
suplub2
Distinct variable groups:   ,,,   ,,,   ,,,   ,
Allowed substitution hints:   (,,)   (,)

Proof of Theorem suplub2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 supmo.1 . . . 4
2 supcl.2 . . . 4
31, 2suplub 7465 . . 3
43expdimp 427 . 2
5 breq2 4216 . . . 4
65cbvrexv 2933 . . 3
7 breq2 4216 . . . . . . 7
87biimprd 215 . . . . . 6
98a1i 11 . . . . 5
101ad2antrr 707 . . . . . . 7
11 simplr 732 . . . . . . 7
12 suplub2.3 . . . . . . . . 9
1312adantr 452 . . . . . . . 8
1413sselda 3348 . . . . . . 7
151, 2supcl 7463 . . . . . . . 8
1615ad2antrr 707 . . . . . . 7
17 sotr 4525 . . . . . . 7
1810, 11, 14, 16, 17syl13anc 1186 . . . . . 6
1918exp3acom23 1381 . . . . 5
201, 2supub 7464 . . . . . . . 8
2120adantr 452 . . . . . . 7
2221imp 419 . . . . . 6
23 sotric 4529 . . . . . . . 8
2410, 16, 14, 23syl12anc 1182 . . . . . . 7
2524con2bid 320 . . . . . 6
2622, 25mpbird 224 . . . . 5
279, 19, 26mpjaod 371 . . . 4
2827rexlimdva 2830 . . 3
296, 28syl5bi 209 . 2
304, 29impbid 184 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359   wceq 1652   wcel 1725  wral 2705  wrex 2706   wss 3320   class class class wbr 4212   wor 4502  csup 7445 This theorem is referenced by:  suprlub  9970  infmrgelb  9988  supxrlub  10904  infmxrgelb  10913  infrglb  27698 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 2417 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-po 4503  df-so 4504  df-iota 5418  df-riota 6549  df-sup 7446
 Copyright terms: Public domain W3C validator