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Theorem somo 4540
 Description: A totally ordered set has at most one minimal element. (Contributed by Mario Carneiro, 24-Jun-2015.) (Revised by NM, 16-Jun-2017.)
Assertion
Ref Expression
somo
Distinct variable groups:   ,,   ,,

Proof of Theorem somo
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 breq1 4218 . . . . . . . . . . 11
21notbid 287 . . . . . . . . . 10
32rspcv 3050 . . . . . . . . 9
4 breq1 4218 . . . . . . . . . . 11
54notbid 287 . . . . . . . . . 10
65rspcv 3050 . . . . . . . . 9
73, 6im2anan9 810 . . . . . . . 8
87ancomsd 442 . . . . . . 7
98imp 420 . . . . . 6
10 ioran 478 . . . . . . 7
11 solin 4529 . . . . . . . . 9
12 df-3or 938 . . . . . . . . . 10
13 or32 515 . . . . . . . . . 10
1412, 13bitri 242 . . . . . . . . 9
1511, 14sylib 190 . . . . . . . 8
1615ord 368 . . . . . . 7
1710, 16syl5bir 211 . . . . . 6
189, 17syl5 31 . . . . 5
1918exp4b 592 . . . 4
2019pm2.43d 47 . . 3
2120ralrimivv 2799 . 2
22 breq2 4219 . . . . 5
2322notbid 287 . . . 4
2423ralbidv 2727 . . 3
2524rmo4 3129 . 2
2621, 25sylibr 205 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 359   wa 360   w3o 936   wcel 1726  wral 2707  wrmo 2710   class class class wbr 4215   wor 4505 This theorem is referenced by:  wereu  4581  wereu2  4582 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rmo 2715  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-so 4507
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