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Theorem somo 4794
 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 4398 . . . . . . . . . . 11
21notbid 301 . . . . . . . . . 10
32rspcv 3132 . . . . . . . . 9
4 breq1 4398 . . . . . . . . . . 11
54notbid 301 . . . . . . . . . 10
65rspcv 3132 . . . . . . . . 9
73, 6im2anan9 853 . . . . . . . 8
87ancomsd 461 . . . . . . 7
98imp 436 . . . . . 6
10 ioran 498 . . . . . . 7
11 solin 4783 . . . . . . . . 9
12 df-3or 1008 . . . . . . . . . 10
13 or32 536 . . . . . . . . . 10
1412, 13bitri 257 . . . . . . . . 9
1511, 14sylib 201 . . . . . . . 8
1615ord 384 . . . . . . 7
1710, 16syl5bir 226 . . . . . 6
189, 17syl5 32 . . . . 5
1918exp4b 618 . . . 4
2019pm2.43d 49 . . 3
2120ralrimivv 2813 . 2
22 breq2 4399 . . . . 5
2322notbid 301 . . . 4
2423ralbidv 2829 . . 3
2524rmo4 3219 . 2
2621, 25sylibr 217 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wo 375   wa 376   w3o 1006   wcel 1904  wral 2756  wrmo 2759   class class class wbr 4395   wor 4759 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rmo 2764  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-so 4761 This theorem is referenced by:  wereu  4835  wereu2  4836
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