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Theorem ordsucss 4488
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 4363 . 2 (Ord 𝐵 → Tr 𝐵)
2 trss 4096 . . . . 5 (Tr 𝐵 → (𝐴𝐵𝐴𝐵))
3 snssi 3724 . . . . . 6 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
43a1i 9 . . . . 5 (Tr 𝐵 → (𝐴𝐵 → {𝐴} ⊆ 𝐵))
52, 4jcad 305 . . . 4 (Tr 𝐵 → (𝐴𝐵 → (𝐴𝐵 ∧ {𝐴} ⊆ 𝐵)))
6 unss 3301 . . . 4 ((𝐴𝐵 ∧ {𝐴} ⊆ 𝐵) ↔ (𝐴 ∪ {𝐴}) ⊆ 𝐵)
75, 6syl6ib 160 . . 3 (Tr 𝐵 → (𝐴𝐵 → (𝐴 ∪ {𝐴}) ⊆ 𝐵))
8 df-suc 4356 . . . 4 suc 𝐴 = (𝐴 ∪ {𝐴})
98sseq1i 3173 . . 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 2141  cun 3119  wss 3121  {csn 3583  Tr wtr 4087  Ord word 4347  suc csuc 4350
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-uni 3797  df-tr 4088  df-iord 4351  df-suc 4356
This theorem is referenced by:  ordelsuc  4489  tfrlemibfn  6307  tfr1onlembfn  6323  tfrcllembfn  6336  sucinc2  6425  nndomo  6842  prarloclemn  7461  ennnfonelemhom  12370  ennnfonelemrn  12374
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