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Theorem ordsucss 4230
 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 → (𝐴𝐵 → suc 𝐴𝐵))

Proof of Theorem ordsucss
StepHypRef Expression
1 ordtr 4115 . 2 (Ord 𝐵 → Tr 𝐵)
2 trss 3863 . . . . 5 (Tr 𝐵 → (𝐴𝐵𝐴𝐵))
3 snssi 3508 . . . . . 6 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
43a1i 9 . . . . 5 (Tr 𝐵 → (𝐴𝐵 → {𝐴} ⊆ 𝐵))
52, 4jcad 291 . . . 4 (Tr 𝐵 → (𝐴𝐵 → (𝐴𝐵 ∧ {𝐴} ⊆ 𝐵)))
6 unss 3117 . . . 4 ((𝐴𝐵 ∧ {𝐴} ⊆ 𝐵) ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵)
75, 6syl6ib 150 . . 3 (Tr 𝐵 → (𝐴𝐵 → (𝐴 ∪ {𝐴}) ⊆ 𝐵))
8 df-suc 4108 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
98sseq1i 2969 . . 3 (suc 𝐴𝐵 ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵)
107, 9syl6ibr 151 . 2 (Tr 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
111, 10syl 14 1 (Ord 𝐵 → (𝐴𝐵 → suc 𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393   ∪ cun 2915   ⊆ wss 2917  {csn 3375  Tr wtr 3854  Ord word 4099  suc csuc 4102 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-uni 3581  df-tr 3855  df-iord 4103  df-suc 4108 This theorem is referenced by:  ordelsuc  4231  tfrlemibfn  5942  sucinc2  6026  nndomo  6326  prarloclemn  6597
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