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Theorem eqsup 7461
 Description: Sufficient condition for an element to be equal to the supremum. (Contributed by Mario Carneiro, 21-Apr-2015.)
Hypothesis
Ref Expression
supmo.1
Assertion
Ref Expression
eqsup
Distinct variable groups:   ,,   ,,   ,,   ,
Allowed substitution hints:   (,)   ()

Proof of Theorem eqsup
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 supmo.1 . . . . 5
21adantr 452 . . . 4
3 simpr1 963 . . . . 5
4 3simpc 956 . . . . . 6
54adantl 453 . . . . 5
6 breq1 4215 . . . . . . . . 9
76notbid 286 . . . . . . . 8
87ralbidv 2725 . . . . . . 7
9 breq2 4216 . . . . . . . . 9
109imbi1d 309 . . . . . . . 8
1110ralbidv 2725 . . . . . . 7
128, 11anbi12d 692 . . . . . 6
1312rspcev 3052 . . . . 5
143, 5, 13syl2anc 643 . . . 4
152, 14supval2 7460 . . 3
162, 14supeu 7459 . . . . 5
1712riota2 6572 . . . . 5
183, 16, 17syl2anc 643 . . . 4
195, 18mpbid 202 . . 3
2015, 19eqtrd 2468 . 2
2120ex 424 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705  wrex 2706  wreu 2707   class class class wbr 4212   wor 4502  crio 6542  csup 7445 This theorem is referenced by:  eqsupd  7462  suprzcl2  10566  supxr  10891 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
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