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Theorem limsuc2 39519
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 7548 . . . . 5 (Ord 𝐴 → (𝐴 = 𝐴 ↔ ∀𝑥𝐴 suc 𝑥𝐴))
21biimpa 477 . . . 4 ((Ord 𝐴𝐴 = 𝐴) → ∀𝑥𝐴 suc 𝑥𝐴)
3 suceq 6249 . . . . . 6 (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵)
43eleq1d 2894 . . . . 5 (𝑥 = 𝐵 → (suc 𝑥𝐴 ↔ suc 𝐵𝐴))
54rspccva 3619 . . . 4 ((∀𝑥𝐴 suc 𝑥𝐴𝐵𝐴) → suc 𝐵𝐴)
62, 5sylan 580 . . 3 (((Ord 𝐴𝐴 = 𝐴) ∧ 𝐵𝐴) → suc 𝐵𝐴)
76ex 413 . 2 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 → suc 𝐵𝐴))
8 ordtr 6198 . . . 4 (Ord 𝐴 → Tr 𝐴)
9 trsuc 6268 . . . . 5 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
109ex 413 . . . 4 (Tr 𝐴 → (suc 𝐵𝐴𝐵𝐴))
118, 10syl 17 . . 3 (Ord 𝐴 → (suc 𝐵𝐴𝐵𝐴))
1211adantr 481 . 2 ((Ord 𝐴𝐴 = 𝐴) → (suc 𝐵𝐴𝐵𝐴))
137, 12impbid 213 1 ((Ord 𝐴𝐴 = 𝐴) → (𝐵𝐴 ↔ suc 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135   cuni 4830  Tr wtr 5163  Ord word 6183  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-suc 6190
This theorem is referenced by:  aomclem4  39535
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