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Theorem ordsucss 4462
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 4338 . 2 (Ord 𝐵 → Tr 𝐵)
2 trss 4071 . . . . 5 (Tr 𝐵 → (𝐴𝐵𝐴𝐵))
3 snssi 3700 . . . . . 6 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
43a1i 9 . . . . 5 (Tr 𝐵 → (𝐴𝐵 → {𝐴} ⊆ 𝐵))
52, 4jcad 305 . . . 4 (Tr 𝐵 → (𝐴𝐵 → (𝐴𝐵 ∧ {𝐴} ⊆ 𝐵)))
6 unss 3281 . . . 4 ((𝐴𝐵 ∧ {𝐴} ⊆ 𝐵) ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵)
75, 6syl6ib 160 . . 3 (Tr 𝐵 → (𝐴𝐵 → (𝐴 ∪ {𝐴}) ⊆ 𝐵))
8 df-suc 4331 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
98sseq1i 3154 . . 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 2128  cun 3100  wss 3102  {csn 3560  Tr wtr 4062  Ord word 4322  suc csuc 4325
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-uni 3773  df-tr 4063  df-iord 4326  df-suc 4331
This theorem is referenced by:  ordelsuc  4463  tfrlemibfn  6272  tfr1onlembfn  6288  tfrcllembfn  6301  sucinc2  6390  nndomo  6806  prarloclemn  7414  ennnfonelemhom  12131  ennnfonelemrn  12135
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