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Theorem ordsucss 4420
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 4300 . 2 (Ord 𝐵 → Tr 𝐵)
2 trss 4035 . . . . 5 (Tr 𝐵 → (𝐴𝐵𝐴𝐵))
3 snssi 3664 . . . . . 6 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
43a1i 9 . . . . 5 (Tr 𝐵 → (𝐴𝐵 → {𝐴} ⊆ 𝐵))
52, 4jcad 305 . . . 4 (Tr 𝐵 → (𝐴𝐵 → (𝐴𝐵 ∧ {𝐴} ⊆ 𝐵)))
6 unss 3250 . . . 4 ((𝐴𝐵 ∧ {𝐴} ⊆ 𝐵) ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵)
75, 6syl6ib 160 . . 3 (Tr 𝐵 → (𝐴𝐵 → (𝐴 ∪ {𝐴}) ⊆ 𝐵))
8 df-suc 4293 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
98sseq1i 3123 . . 3 (suc 𝐴𝐵 ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵)
107, 9syl6ibr 161 . 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 103  wcel 1480  cun 3069  wss 3071  {csn 3527  Tr wtr 4026  Ord word 4284  suc csuc 4287
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-uni 3737  df-tr 4027  df-iord 4288  df-suc 4293
This theorem is referenced by:  ordelsuc  4421  tfrlemibfn  6225  tfr1onlembfn  6241  tfrcllembfn  6254  sucinc2  6342  nndomo  6758  prarloclemn  7314  ennnfonelemhom  11935  ennnfonelemrn  11939
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