Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trintALT Structured version   Unicode version

Theorem trintALT 32638
Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 32638 is an alternative proof of trint 4550. trintALT 32638 is trintALTVD 32637 without virtual deductions and was automatically derived from trintALTVD 32637 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trintALT  |-  ( A. x  e.  A  Tr  x  ->  Tr  |^| A )
Distinct variable group:    x, A

Proof of Theorem trintALT
Dummy variables  q 
y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . . 5  |-  ( ( z  e.  y  /\  y  e.  |^| A )  ->  z  e.  y )
21a1i 11 . . . 4  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  -> 
z  e.  y ) )
3 iidn3 32226 . . . . . . 7  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  -> 
( q  e.  A  ->  q  e.  A ) ) )
4 id 22 . . . . . . . 8  |-  ( A. x  e.  A  Tr  x  ->  A. x  e.  A  Tr  x )
5 rspsbc 3416 . . . . . . . 8  |-  ( q  e.  A  ->  ( A. x  e.  A  Tr  x  ->  [. q  /  x ]. Tr  x
) )
63, 4, 5ee31 32506 . . . . . . 7  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  -> 
( q  e.  A  ->  [. q  /  x ]. Tr  x ) ) )
7 trsbc 32268 . . . . . . . 8  |-  ( q  e.  A  ->  ( [. q  /  x ]. Tr  x  <->  Tr  q
) )
87biimpd 207 . . . . . . 7  |-  ( q  e.  A  ->  ( [. q  /  x ]. Tr  x  ->  Tr  q ) )
93, 6, 8ee33 32247 . . . . . 6  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  -> 
( q  e.  A  ->  Tr  q ) ) )
10 simpr 461 . . . . . . . . 9  |-  ( ( z  e.  y  /\  y  e.  |^| A )  ->  y  e.  |^| A )
1110a1i 11 . . . . . . . 8  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  -> 
y  e.  |^| A
) )
12 elintg 4285 . . . . . . . . 9  |-  ( y  e.  |^| A  ->  (
y  e.  |^| A  <->  A. q  e.  A  y  e.  q ) )
1312ibi 241 . . . . . . . 8  |-  ( y  e.  |^| A  ->  A. q  e.  A  y  e.  q )
1411, 13syl6 33 . . . . . . 7  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  ->  A. q  e.  A  y  e.  q )
)
15 rsp 2825 . . . . . . 7  |-  ( A. q  e.  A  y  e.  q  ->  ( q  e.  A  ->  y  e.  q ) )
1614, 15syl6 33 . . . . . 6  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  -> 
( q  e.  A  ->  y  e.  q ) ) )
17 trel 4542 . . . . . . 7  |-  ( Tr  q  ->  ( (
z  e.  y  /\  y  e.  q )  ->  z  e.  q ) )
1817expd 436 . . . . . 6  |-  ( Tr  q  ->  ( z  e.  y  ->  ( y  e.  q  ->  z  e.  q ) ) )
199, 2, 16, 18ee323 32233 . . . . 5  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  -> 
( q  e.  A  ->  z  e.  q ) ) )
2019ralrimdv 2875 . . . 4  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  ->  A. q  e.  A  z  e.  q )
)
21 elintg 4285 . . . . 5  |-  ( z  e.  y  ->  (
z  e.  |^| A  <->  A. q  e.  A  z  e.  q ) )
2221biimprd 223 . . . 4  |-  ( z  e.  y  ->  ( A. q  e.  A  z  e.  q  ->  z  e.  |^| A ) )
232, 20, 22syl6c 64 . . 3  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  -> 
z  e.  |^| A
) )
2423alrimivv 1691 . 2  |-  ( A. x  e.  A  Tr  x  ->  A. z A. y
( ( z  e.  y  /\  y  e. 
|^| A )  -> 
z  e.  |^| A
) )
25 dftr2 4537 . 2  |-  ( Tr 
|^| A  <->  A. z A. y ( ( z  e.  y  /\  y  e.  |^| A )  -> 
z  e.  |^| A
) )
2624, 25sylibr 212 1  |-  ( A. x  e.  A  Tr  x  ->  Tr  |^| A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1372    e. wcel 1762   A.wral 2809   [.wsbc 3326   |^|cint 4277   Tr wtr 4535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-v 3110  df-sbc 3327  df-in 3478  df-ss 3485  df-uni 4241  df-int 4278  df-tr 4536
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator