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Theorem trss 3891
Description: An element of a transitive class is a subset of the class. (Contributed by NM, 7-Aug-1994.)
Assertion
Ref Expression
trss (Tr 𝐴 → (𝐵𝐴𝐵𝐴))

Proof of Theorem trss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2116 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
2 sseq1 2994 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
31, 2imbi12d 227 . . . 4 (𝑥 = 𝐵 → ((𝑥𝐴𝑥𝐴) ↔ (𝐵𝐴𝐵𝐴)))
43imbi2d 223 . . 3 (𝑥 = 𝐵 → ((Tr 𝐴 → (𝑥𝐴𝑥𝐴)) ↔ (Tr 𝐴 → (𝐵𝐴𝐵𝐴))))
5 dftr3 3886 . . . 4 (Tr 𝐴 ↔ ∀𝑥𝐴 𝑥𝐴)
6 rsp 2386 . . . 4 (∀𝑥𝐴 𝑥𝐴 → (𝑥𝐴𝑥𝐴))
75, 6sylbi 118 . . 3 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
84, 7vtoclg 2630 . 2 (𝐵𝐴 → (Tr 𝐴 → (𝐵𝐴𝐵𝐴)))
98pm2.43b 50 1 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  wral 2323  wss 2945  Tr wtr 3882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-tr 3883
This theorem is referenced by:  trin  3892  triun  3895  trintssm  3898  tz7.2  4119  ordelss  4144  trsucss  4188  ordsucss  4258
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