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Theorem ordsucss 4196
 Description: The successor of an element of an ordinal class is a subset of it. (Contributed by NM, 21-Jun-1998.)
Assertion
Ref Expression
ordsucss (Ord B → (A B → suc AB))

Proof of Theorem ordsucss
StepHypRef Expression
1 ordtr 4081 . 2 (Ord B → Tr B)
2 trss 3854 . . . . 5 (Tr B → (A BAB))
3 snssi 3499 . . . . . 6 (A B → {A} ⊆ B)
43a1i 9 . . . . 5 (Tr B → (A B → {A} ⊆ B))
52, 4jcad 291 . . . 4 (Tr B → (A B → (AB {A} ⊆ B)))
6 unss 3111 . . . 4 ((AB {A} ⊆ B) ↔ (A ∪ {A}) ⊆ B)
75, 6syl6ib 150 . . 3 (Tr B → (A B → (A ∪ {A}) ⊆ B))
8 df-suc 4074 . . . 4 suc A = (A ∪ {A})
98sseq1i 2963 . . 3 (suc AB ↔ (A ∪ {A}) ⊆ B)
107, 9syl6ibr 151 . 2 (Tr B → (A B → suc AB))
111, 10syl 14 1 (Ord B → (A B → suc AB))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390   ∪ cun 2909   ⊆ wss 2911  {csn 3367  Tr wtr 3845  Ord word 4065  suc csuc 4068 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-uni 3572  df-tr 3846  df-iord 4069  df-suc 4074 This theorem is referenced by:  ordelsuc  4197  tfrlemibfn  5883  sucinc2  5965  prarloclemn  6481
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