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Theorem trsucss 4274
Description: A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
Assertion
Ref Expression
trsucss (Tr 𝐴 → (𝐵 ∈ suc 𝐴𝐵𝐴))

Proof of Theorem trsucss
StepHypRef Expression
1 elsuci 4254 . 2 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
2 trss 3967 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
3 eqimss 3093 . . . 4 (𝐵 = 𝐴𝐵𝐴)
43a1i 9 . . 3 (Tr 𝐴 → (𝐵 = 𝐴𝐵𝐴))
52, 4jaod 675 . 2 (Tr 𝐴 → ((𝐵𝐴𝐵 = 𝐴) → 𝐵𝐴))
61, 5syl5 32 1 (Tr 𝐴 → (𝐵 ∈ suc 𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 667   = wceq 1296  wcel 1445  wss 3013  Tr wtr 3958  suc csuc 4216
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-un 3017  df-in 3019  df-ss 3026  df-sn 3472  df-uni 3676  df-tr 3959  df-suc 4222
This theorem is referenced by:  onsucsssucr  4354  ordpwsucss  4411  nnsf  12600  nninfalllemn  12603
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