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Theorem soss 4042
Description: Subset theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
soss (AB → (𝑅 Or B𝑅 Or A))

Proof of Theorem soss
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poss 4026 . . 3 (AB → (𝑅 Po B𝑅 Po A))
2 ssel 2933 . . . . . . . 8 (AB → (x Ax B))
3 ssel 2933 . . . . . . . 8 (AB → (y Ay B))
4 ssel 2933 . . . . . . . 8 (AB → (z Az B))
52, 3, 43anim123d 1213 . . . . . . 7 (AB → ((x A y A z A) → (x B y B z B)))
65imim1d 69 . . . . . 6 (AB → (((x B y B z B) → (x𝑅y → (x𝑅z z𝑅y))) → ((x A y A z A) → (x𝑅y → (x𝑅z z𝑅y)))))
762alimdv 1758 . . . . 5 (AB → (yz((x B y B z B) → (x𝑅y → (x𝑅z z𝑅y))) → yz((x A y A z A) → (x𝑅y → (x𝑅z z𝑅y)))))
87alimdv 1756 . . . 4 (AB → (xyz((x B y B z B) → (x𝑅y → (x𝑅z z𝑅y))) → xyz((x A y A z A) → (x𝑅y → (x𝑅z z𝑅y)))))
9 r3al 2360 . . . 4 (x B y B z B (x𝑅y → (x𝑅z z𝑅y)) ↔ xyz((x B y B z B) → (x𝑅y → (x𝑅z z𝑅y))))
10 r3al 2360 . . . 4 (x A y A z A (x𝑅y → (x𝑅z z𝑅y)) ↔ xyz((x A y A z A) → (x𝑅y → (x𝑅z z𝑅y))))
118, 9, 103imtr4g 194 . . 3 (AB → (x B y B z B (x𝑅y → (x𝑅z z𝑅y)) → x A y A z A (x𝑅y → (x𝑅z z𝑅y))))
121, 11anim12d 318 . 2 (AB → ((𝑅 Po B x B y B z B (x𝑅y → (x𝑅z z𝑅y))) → (𝑅 Po A x A y A z A (x𝑅y → (x𝑅z z𝑅y)))))
13 df-iso 4025 . 2 (𝑅 Or B ↔ (𝑅 Po B x B y B z B (x𝑅y → (x𝑅z z𝑅y))))
14 df-iso 4025 . 2 (𝑅 Or A ↔ (𝑅 Po A x A y A z A (x𝑅y → (x𝑅z z𝑅y))))
1512, 13, 143imtr4g 194 1 (AB → (𝑅 Or B𝑅 Or A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 628   w3a 884  wal 1240   wcel 1390  wral 2300  wss 2911   class class class wbr 3755   Po wpo 4022   Or wor 4023
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  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-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-in 2918  df-ss 2925  df-po 4024  df-iso 4025
This theorem is referenced by:  soeq2  4044
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