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Theorem sess1 4059
Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
sess1 (𝑅𝑆 → (𝑆 Se A𝑅 Se A))

Proof of Theorem sess1
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 102 . . . . . 6 ((𝑅𝑆 y A) → 𝑅𝑆)
21ssbrd 3796 . . . . 5 ((𝑅𝑆 y A) → (y𝑅xy𝑆x))
32ss2rabdv 3015 . . . 4 (𝑅𝑆 → {y Ay𝑅x} ⊆ {y Ay𝑆x})
4 ssexg 3887 . . . . 5 (({y Ay𝑅x} ⊆ {y Ay𝑆x} {y Ay𝑆x} V) → {y Ay𝑅x} V)
54ex 108 . . . 4 ({y Ay𝑅x} ⊆ {y Ay𝑆x} → ({y Ay𝑆x} V → {y Ay𝑅x} V))
63, 5syl 14 . . 3 (𝑅𝑆 → ({y Ay𝑆x} V → {y Ay𝑅x} V))
76ralimdv 2382 . 2 (𝑅𝑆 → (x A {y Ay𝑆x} V → x A {y Ay𝑅x} V))
8 df-se 4056 . 2 (𝑆 Se Ax A {y Ay𝑆x} V)
9 df-se 4056 . 2 (𝑅 Se Ax A {y Ay𝑅x} V)
107, 8, 93imtr4g 194 1 (𝑅𝑆 → (𝑆 Se A𝑅 Se A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  wral 2300  {crab 2304  Vcvv 2551  wss 2911   class class class wbr 3755   Se wse 4055
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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866
This theorem depends on definitions:  df-bi 110  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-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-br 3756  df-se 4056
This theorem is referenced by:  seeq1  4061
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