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Theorem trintALT 28930
 Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 28930 is an alternative proof of trint 4309. trintALT 28930 is trintALTVD 28929 without virtual deductions and was automatically derived from trintALTVD 28929 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
Distinct variable group:   ,

Proof of Theorem trintALT
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 444 . . . . 5
21a1i 11 . . . 4
3 iidn3 28520 . . . . . . 7
4 id 20 . . . . . . . 8
5 rspsbc 3231 . . . . . . . 8
63, 4, 5ee31 28801 . . . . . . 7
7 trsbc 28562 . . . . . . . 8
87biimpd 199 . . . . . . 7
93, 6, 8ee33 28542 . . . . . 6
10 simpr 448 . . . . . . . . 9
1110a1i 11 . . . . . . . 8
12 elintg 4050 . . . . . . . . 9
1312ibi 233 . . . . . . . 8
1411, 13syl6 31 . . . . . . 7
15 rsp 2758 . . . . . . 7
1614, 15syl6 31 . . . . . 6
17 trel 4301 . . . . . . 7
1817exp3a 426 . . . . . 6
199, 2, 16, 18ee323 28527 . . . . 5
2019ralrimdv 2787 . . . 4
21 elintg 4050 . . . . 5
2221biimprd 215 . . . 4
232, 20, 22ee22 1371 . . 3
2423alrimivv 1642 . 2
25 dftr2 4296 . 2
2624, 25sylibr 204 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549   wcel 1725  wral 2697  wsbc 3153  cint 4042   wtr 4294 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-v 2950  df-sbc 3154  df-in 3319  df-ss 3326  df-uni 4008  df-int 4043  df-tr 4295
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