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Theorem trsucss 4398
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 4378 . 2 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
2 trss 4086 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
3 eqimss 3194 . . . 4 (𝐵 = 𝐴𝐵𝐴)
43a1i 9 . . 3 (Tr 𝐴 → (𝐵 = 𝐴𝐵𝐴))
52, 4jaod 707 . 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 698   = wceq 1342  wcel 2135  wss 3114  Tr wtr 4077  suc csuc 4340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-v 2726  df-un 3118  df-in 3120  df-ss 3127  df-sn 3579  df-uni 3787  df-tr 4078  df-suc 4346
This theorem is referenced by:  onsucsssucr  4483  ordpwsucss  4541  nnnninfeq  7086  bj-el2oss1o  13548  nnsf  13778
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