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Theorem sosng 4413
 Description: Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
Assertion
Ref Expression
sosng

Proof of Theorem sosng
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sopo 4050 . . 3
2 posng 4412 . . 3
31, 2syl5ib 143 . 2
42biimpar 281 . . . 4
5 ax-in2 545 . . . . . . . . 9
65adantr 261 . . . . . . . 8
7 elsni 3393 . . . . . . . . . . 11
8 elsni 3393 . . . . . . . . . . 11
97, 8breqan12d 3779 . . . . . . . . . 10
109imbi1d 220 . . . . . . . . 9
1110adantl 262 . . . . . . . 8
126, 11mpbird 156 . . . . . . 7
1312ralrimivw 2393 . . . . . 6
1413ralrimivva 2401 . . . . 5
1514adantl 262 . . . 4
16 df-iso 4034 . . . 4
174, 15, 16sylanbrc 394 . . 3
1817ex 108 . 2
193, 18impbid 120 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 629   wcel 1393  wral 2306  cvv 2557  csn 3375   class class class wbr 3764   wpo 4031   wor 4032   wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-sbc 2765  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-po 4033  df-iso 4034 This theorem is referenced by: (None)
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