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Theorem trintALT 27347
 Description: The intersection of a class of transitive sets is transitive. Exercise 5(b) of [Enderton] p. 73. trintALT 27347 is an alternative proof of trint 4025. trintALT 27347 is trintALTVD 27346 without virtual deductions and was automatically derived from trintALTVD 27346 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
StepHypRef Expression
1 simpl 445 . . . . 5
21a1i 12 . . . 4
3 iidn3 26955 . . . . . . 7
4 id 21 . . . . . . . 8
5 ra4sbc 2999 . . . . . . . 8
63, 4, 5ee31 27217 . . . . . . 7
7 trsbc 26997 . . . . . . . 8
87biimpd 200 . . . . . . 7
93, 6, 8ee33 26977 . . . . . 6
10 simpr 449 . . . . . . . . 9
1110a1i 12 . . . . . . . 8
12 elintg 3768 . . . . . . . . 9
1312ibi 234 . . . . . . . 8
1411, 13syl6 31 . . . . . . 7
15 ra4 2565 . . . . . . 7
1614, 15syl6 31 . . . . . 6
17 trel 4017 . . . . . . 7
1817exp3a 427 . . . . . 6
199, 2, 16, 18ee323 26962 . . . . 5
2019ralrimdv 2594 . . . 4
21 elintg 3768 . . . . 5
2221biimprd 216 . . . 4
232, 20, 22ee22 1358 . . 3
2423alrimivv 2013 . 2
25 dftr2 4012 . 2
2624, 25sylibr 205 1
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360  wal 1532   wcel 1621  wral 2509  wsbc 2921  cint 3760   wtr 4010 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ral 2513  df-v 2729  df-sbc 2922  df-in 3085  df-ss 3089  df-uni 3728  df-int 3761  df-tr 4011
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