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Theorem trelpss 40942
 Description: An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 5515, ax-reg 9034 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)
Assertion
Ref Expression
trelpss ((Tr 𝐴𝐵𝐴) → 𝐵𝐴)

Proof of Theorem trelpss
StepHypRef Expression
1 zfregfr 9046 . . 3 E Fr 𝐴
2 tz7.2 5515 . . 3 ((Tr 𝐴 ∧ E Fr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))
31, 2mp3an2 1445 . 2 ((Tr 𝐴𝐵𝐴) → (𝐵𝐴𝐵𝐴))
4 df-pss 3932 . 2 (𝐵𝐴 ↔ (𝐵𝐴𝐵𝐴))
53, 4sylibr 236 1 ((Tr 𝐴𝐵𝐴) → 𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2114   ≠ wne 3006   ⊆ wss 3913   ⊊ wpss 3914  Tr wtr 5148   E cep 5440   Fr wfr 5487 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pr 5306  ax-reg 9034 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-tr 5149  df-eprel 5441  df-fr 5490 This theorem is referenced by: (None)
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