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Theorem poss 4055
 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 3059 . . 3
2 ssralv 3059 . . . . 5
3 ssralv 3059 . . . . . 6
43ralimdv 2431 . . . . 5
52, 4syld 44 . . . 4
65ralimdv 2431 . . 3
71, 6syld 44 . 2
8 df-po 4053 . 2
9 df-po 4053 . 2
107, 8, 93imtr4g 203 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 102  wral 2349   wss 2974   class class class wbr 3787   wpo 4051 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-ral 2354  df-in 2980  df-ss 2987  df-po 4053 This theorem is referenced by:  poeq2  4057  soss  4071
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