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Theorem poss 4026
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 (AB → (𝑅 Po B𝑅 Po A))

Proof of Theorem poss
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssralv 2998 . . 3 (AB → (x B y B z Bx𝑅x ((x𝑅y y𝑅z) → x𝑅z)) → x A y B z Bx𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
2 ssralv 2998 . . . . 5 (AB → (y B z Bx𝑅x ((x𝑅y y𝑅z) → x𝑅z)) → y A z Bx𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
3 ssralv 2998 . . . . . 6 (AB → (z Bx𝑅x ((x𝑅y y𝑅z) → x𝑅z)) → z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
43ralimdv 2382 . . . . 5 (AB → (y A z Bx𝑅x ((x𝑅y y𝑅z) → x𝑅z)) → y A z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
52, 4syld 40 . . . 4 (AB → (y B z Bx𝑅x ((x𝑅y y𝑅z) → x𝑅z)) → y A z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
65ralimdv 2382 . . 3 (AB → (x A y B z Bx𝑅x ((x𝑅y y𝑅z) → x𝑅z)) → x A y A z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
71, 6syld 40 . 2 (AB → (x B y B z Bx𝑅x ((x𝑅y y𝑅z) → x𝑅z)) → x A y A z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z))))
8 df-po 4024 . 2 (𝑅 Po Bx B y B z Bx𝑅x ((x𝑅y y𝑅z) → x𝑅z)))
9 df-po 4024 . 2 (𝑅 Po Ax A y A z Ax𝑅x ((x𝑅y y𝑅z) → x𝑅z)))
107, 8, 93imtr4g 194 1 (AB → (𝑅 Po B𝑅 Po A))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wral 2300  wss 2911   class class class wbr 3755   Po wpo 4022
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-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ral 2305  df-in 2918  df-ss 2925  df-po 4024
This theorem is referenced by:  poeq2  4028  soss  4042
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