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Theorem limsuc2 43030
Description: Limit ordinals in the sense inclusive of zero contain all successors of their members. (Contributed by Stefan O'Rear, 20-Jan-2015.)
Assertion
Ref Expression
limsuc2 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))

Proof of Theorem limsuc2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ordunisuc2 7865 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
21biimpa 476 . . . 4 ((Ord 𝐴𝐴 = 𝐴) → ∀𝑥𝐴 suc 𝑥𝐴)
3 suceq 6452 . . . . . 6 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
43eleq1d 2824 . . . . 5 (𝑥 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝐵𝐴))
54rspccva 3621 . . . 4 ((∀𝑥𝐴 suc 𝑥𝐴𝐵𝐴) → suc 𝐵𝐴)
62, 5sylan 580 . . 3 (((Ord 𝐴𝐴 = 𝐴) ∧ 𝐵𝐴) → suc 𝐵𝐴)
76ex 412 . 2 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 → suc 𝐵𝐴))
8 ordtr 6400 . . . 4 (Ord 𝐴 → Tr 𝐴)
9 trsuc 6473 . . . . 5 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
109ex 412 . . . 4 (Tr 𝐴 → (suc 𝐵𝐴𝐵𝐴))
118, 10syl 17 . . 3 (Ord 𝐴 → (suc 𝐵𝐴𝐵𝐴))
1211adantr 480 . 2 ((Ord 𝐴𝐴 = 𝐴) → (suc 𝐵𝐴𝐵𝐴))
137, 12impbid 212 1 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059   cuni 4912  Tr wtr 5265  Ord word 6385  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by:  aomclem4  43046
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