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Theorem limsssuc 7570
 Description: A class includes a limit ordinal iff the successor of the class includes it. (Contributed by NM, 30-Oct-2003.)
Assertion
Ref Expression
limsssuc (Lim 𝐴 → (𝐴𝐵𝐴 ⊆ suc 𝐵))

Proof of Theorem limsssuc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sssucid 6251 . . 3 𝐵 ⊆ suc 𝐵
2 sstr2 3901 . . 3 (𝐴𝐵 → (𝐵 ⊆ suc 𝐵𝐴 ⊆ suc 𝐵))
31, 2mpi 20 . 2 (𝐴𝐵𝐴 ⊆ suc 𝐵)
4 eleq1 2839 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
54biimpcd 252 . . . . . . . . . . 11 (𝑥𝐴 → (𝑥 = 𝐵𝐵𝐴))
6 limsuc 7569 . . . . . . . . . . . . . 14 (Lim 𝐴 → (𝐵𝐴 ↔ suc 𝐵𝐴))
76biimpa 480 . . . . . . . . . . . . 13 ((Lim 𝐴𝐵𝐴) → suc 𝐵𝐴)
8 limord 6233 . . . . . . . . . . . . . . 15 (Lim 𝐴 → Ord 𝐴)
9 ordelord 6196 . . . . . . . . . . . . . . . . 17 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
108, 9sylan 583 . . . . . . . . . . . . . . . 16 ((Lim 𝐴𝐵𝐴) → Ord 𝐵)
11 ordsuc 7534 . . . . . . . . . . . . . . . 16 (Ord 𝐵 ↔ Ord suc 𝐵)
1210, 11sylib 221 . . . . . . . . . . . . . . 15 ((Lim 𝐴𝐵𝐴) → Ord suc 𝐵)
13 ordtri1 6207 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵𝐴))
148, 12, 13syl2an2r 684 . . . . . . . . . . . . . 14 ((Lim 𝐴𝐵𝐴) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵𝐴))
1514con2bid 358 . . . . . . . . . . . . 13 ((Lim 𝐴𝐵𝐴) → (suc 𝐵𝐴 ↔ ¬ 𝐴 ⊆ suc 𝐵))
167, 15mpbid 235 . . . . . . . . . . . 12 ((Lim 𝐴𝐵𝐴) → ¬ 𝐴 ⊆ suc 𝐵)
1716ex 416 . . . . . . . . . . 11 (Lim 𝐴 → (𝐵𝐴 → ¬ 𝐴 ⊆ suc 𝐵))
185, 17sylan9r 512 . . . . . . . . . 10 ((Lim 𝐴𝑥𝐴) → (𝑥 = 𝐵 → ¬ 𝐴 ⊆ suc 𝐵))
1918con2d 136 . . . . . . . . 9 ((Lim 𝐴𝑥𝐴) → (𝐴 ⊆ suc 𝐵 → ¬ 𝑥 = 𝐵))
2019ex 416 . . . . . . . 8 (Lim 𝐴 → (𝑥𝐴 → (𝐴 ⊆ suc 𝐵 → ¬ 𝑥 = 𝐵)))
2120com23 86 . . . . . . 7 (Lim 𝐴 → (𝐴 ⊆ suc 𝐵 → (𝑥𝐴 → ¬ 𝑥 = 𝐵)))
2221imp31 421 . . . . . 6 (((Lim 𝐴𝐴 ⊆ suc 𝐵) ∧ 𝑥𝐴) → ¬ 𝑥 = 𝐵)
23 ssel2 3889 . . . . . . . . . 10 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → 𝑥 ∈ suc 𝐵)
24 vex 3413 . . . . . . . . . . 11 𝑥 ∈ V
2524elsuc 6243 . . . . . . . . . 10 (𝑥 ∈ suc 𝐵 ↔ (𝑥𝐵𝑥 = 𝐵))
2623, 25sylib 221 . . . . . . . . 9 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → (𝑥𝐵𝑥 = 𝐵))
2726ord 861 . . . . . . . 8 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → (¬ 𝑥𝐵𝑥 = 𝐵))
2827con1d 147 . . . . . . 7 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → (¬ 𝑥 = 𝐵𝑥𝐵))
2928adantll 713 . . . . . 6 (((Lim 𝐴𝐴 ⊆ suc 𝐵) ∧ 𝑥𝐴) → (¬ 𝑥 = 𝐵𝑥𝐵))
3022, 29mpd 15 . . . . 5 (((Lim 𝐴𝐴 ⊆ suc 𝐵) ∧ 𝑥𝐴) → 𝑥𝐵)
3130ex 416 . . . 4 ((Lim 𝐴𝐴 ⊆ suc 𝐵) → (𝑥𝐴𝑥𝐵))
3231ssrdv 3900 . . 3 ((Lim 𝐴𝐴 ⊆ suc 𝐵) → 𝐴𝐵)
3332ex 416 . 2 (Lim 𝐴 → (𝐴 ⊆ suc 𝐵𝐴𝐵))
343, 33impbid2 229 1 (Lim 𝐴 → (𝐴𝐵𝐴 ⊆ suc 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ⊆ wss 3860  Ord word 6173  Lim wlim 6175  suc csuc 6176 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-tr 5143  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180 This theorem is referenced by:  cardlim  9447
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