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Theorem trelpss 27636
 Description: An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 4566, ax-reg 7560 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)
Assertion
Ref Expression
trelpss

Proof of Theorem trelpss
StepHypRef Expression
1 zfregfr 7570 . . 3
2 tz7.2 4566 . . 3
31, 2mp3an2 1267 . 2
4 df-pss 3336 . 2
53, 4sylibr 204 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wcel 1725   wne 2599   wss 3320   wpss 3321   wtr 4302   cep 4492   wfr 4538 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-reg 7560 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-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-fr 4541
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