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Theorem ordsucss 4390
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 4270 . 2 (Ord 𝐵 → Tr 𝐵)
2 trss 4005 . . . . 5 (Tr 𝐵 → (𝐴𝐵𝐴𝐵))
3 snssi 3634 . . . . . 6 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
43a1i 9 . . . . 5 (Tr 𝐵 → (𝐴𝐵 → {𝐴} ⊆ 𝐵))
52, 4jcad 305 . . . 4 (Tr 𝐵 → (𝐴𝐵 → (𝐴𝐵 ∧ {𝐴} ⊆ 𝐵)))
6 unss 3220 . . . 4 ((𝐴𝐵 ∧ {𝐴} ⊆ 𝐵) ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵)
75, 6syl6ib 160 . . 3 (Tr 𝐵 → (𝐴𝐵 → (𝐴 ∪ {𝐴}) ⊆ 𝐵))
8 df-suc 4263 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
98sseq1i 3093 . . 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 1465  cun 3039  wss 3041  {csn 3497  Tr wtr 3996  Ord word 4254  suc csuc 4257
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-uni 3707  df-tr 3997  df-iord 4258  df-suc 4263
This theorem is referenced by:  ordelsuc  4391  tfrlemibfn  6193  tfr1onlembfn  6209  tfrcllembfn  6222  sucinc2  6310  nndomo  6726  prarloclemn  7275  ennnfonelemhom  11855  ennnfonelemrn  11859
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