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Theorem poss 4220
 Description: Subset theorem for the partial ordering predicate. (Contributed by NM, 27-Mar-1997.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
poss

Proof of Theorem poss
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssralv 3161 . . 3
2 ssralv 3161 . . . . 5
3 ssralv 3161 . . . . . 6
43ralimdv 2500 . . . . 5
52, 4syld 45 . . . 4
65ralimdv 2500 . . 3
71, 6syld 45 . 2
8 df-po 4218 . 2
9 df-po 4218 . 2
107, 8, 93imtr4g 204 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103  wral 2416   wss 3071   class class class wbr 3929   wpo 4216 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-ral 2421  df-in 3077  df-ss 3084  df-po 4218 This theorem is referenced by:  poeq2  4222  soss  4236  fimaxq  10580
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