Theorem bnj72 13208

Description: Property of Se.

Assertion

Ref	Expression
bnj72	|- ((R Se A /\ B C_ A) -> R Se B)

Proof of Theorem bnj72

Step	Hyp	Ref	Expression
1		bnj73 13207	. . . . . . 7 |- (B C_ A -> pred(x, B, R) C_ pred(x, A, R))
2		bnj74 12435	. . . . . . 7 |- ( pred(x, B, R) C_ pred(x, A, R) -> ( pred(x, A, R) e. _V -> pred(x, B, R) e. _V))
3	1, 2	syl 12	. . . . . 6 |- (B C_ A -> ( pred(x, A, R) e. _V -> pred(x, B, R) e. _V))
4	3	19.21aiv 1664	. . . . 5 |- (B C_ A -> A.x( pred(x, A, R) e. _V -> pred(x, B, R) e. _V))
5		alral 2153	. . . . 5 |- (A.x( pred(x, A, R) e. _V -> pred(x, B, R) e. _V) -> A.x e. A ( pred(x, A, R) e. _V -> pred(x, B, R) e. _V))
6		ralim 2164	. . . . 5 |- (A.x e. A ( pred(x, A, R) e. _V -> pred(x, B, R) e. _V) -> (A.x e. A pred(x, A, R) e. _V -> A.x e. A pred(x, B, R) e. _V))
7	4, 5, 6	3syl 24	. . . 4 |- (B C_ A -> (A.x e. A pred(x, A, R) e. _V -> A.x e. A pred(x, B, R) e. _V))
8		ssralv 2672	. . . 4 |- (B C_ A -> (A.x e. A pred(x, B, R) e. _V -> A.x e. B pred(x, B, R) e. _V))
9	7, 8	syld 30	. . 3 |- (B C_ A -> (A.x e. A pred(x, A, R) e. _V -> A.x e. B pred(x, B, R) e. _V))
10		df-bnj13 12026	. . 3 |- (R Se A <-> A.x e. A pred(x, A, R) e. _V)
11		df-bnj13 12026	. . 3 |- (R Se B <-> A.x e. B pred(x, B, R) e. _V)
12	9, 10, 11	3imtr4g 612	. 2 |- (B C_ A -> (R Se A -> R Se B))
13	12	impcom 378	1 |- ((R Se A /\ B C_ A) -> R Se B)

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593   predsyn-bnj14 12023   Se syn-bnj13 12025

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rab 2112  df-v 2294  df-in 2603  df-ss 2605  df-bnj14 12024  df-bnj13 12026

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