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Theorem trintALT 33782
Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 33782 is an alternative proof of trint 4565. trintALT 33782 is trintALTVD 33781 without virtual deductions and was automatically derived from trintALTVD 33781 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 33371 . . . . . . 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 3413 . . . . . . . 8  |-  ( q  e.  A  ->  ( A. x  e.  A  Tr  x  ->  [. q  /  x ]. Tr  x
) )
63, 4, 5ee31 33650 . . . . . . 7  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  -> 
( q  e.  A  ->  [. q  /  x ]. Tr  x ) ) )
7 trsbc 33412 . . . . . . . 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 33392 . . . . . 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 4296 . . . . . . . . 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 2823 . . . . . . 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 4557 . . . . . . 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 33378 . . . . 5  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  -> 
( q  e.  A  ->  z  e.  q ) ) )
2019ralrimdv 2873 . . . 4  |-  ( A. x  e.  A  Tr  x  ->  ( ( z  e.  y  /\  y  e.  |^| A )  ->  A. q  e.  A  z  e.  q )
)
21 elintg 4296 . . . . 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 1721 . 2  |-  ( A. x  e.  A  Tr  x  ->  A. z A. y
( ( z  e.  y  /\  y  e. 
|^| A )  -> 
z  e.  |^| A
) )
25 dftr2 4552 . 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 1393    e. wcel 1819   A.wral 2807   [.wsbc 3327   |^|cint 4288   Tr wtr 4550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-sbc 3328  df-in 3478  df-ss 3485  df-uni 4252  df-int 4289  df-tr 4551
This theorem is referenced by: (None)
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