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Theorem limsssuc 7870
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 6465 . . 3 𝐵 ⊆ suc 𝐵
2 sstr2 4001 . . 3 (𝐴𝐵 → (𝐵 ⊆ suc 𝐵𝐴 ⊆ suc 𝐵))
31, 2mpi 20 . 2 (𝐴𝐵𝐴 ⊆ suc 𝐵)
4 eleq1 2826 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
54biimpcd 249 . . . . . . . . . . 11 (𝑥𝐴 → (𝑥 = 𝐵𝐵𝐴))
6 limsuc 7869 . . . . . . . . . . . . . 14 (Lim 𝐴 → (𝐵𝐴 ↔ suc 𝐵𝐴))
76biimpa 476 . . . . . . . . . . . . 13 ((Lim 𝐴𝐵𝐴) → suc 𝐵𝐴)
8 limord 6445 . . . . . . . . . . . . . . 15 (Lim 𝐴 → Ord 𝐴)
9 ordelord 6407 . . . . . . . . . . . . . . . . 17 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
108, 9sylan 580 . . . . . . . . . . . . . . . 16 ((Lim 𝐴𝐵𝐴) → Ord 𝐵)
11 ordsuc 7832 . . . . . . . . . . . . . . . 16 (Ord 𝐵 ↔ Ord suc 𝐵)
1210, 11sylib 218 . . . . . . . . . . . . . . 15 ((Lim 𝐴𝐵𝐴) → Ord suc 𝐵)
13 ordtri1 6418 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵𝐴))
148, 12, 13syl2an2r 685 . . . . . . . . . . . . . 14 ((Lim 𝐴𝐵𝐴) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵𝐴))
1514con2bid 354 . . . . . . . . . . . . 13 ((Lim 𝐴𝐵𝐴) → (suc 𝐵𝐴 ↔ ¬ 𝐴 ⊆ suc 𝐵))
167, 15mpbid 232 . . . . . . . . . . . 12 ((Lim 𝐴𝐵𝐴) → ¬ 𝐴 ⊆ suc 𝐵)
1716ex 412 . . . . . . . . . . 11 (Lim 𝐴 → (𝐵𝐴 → ¬ 𝐴 ⊆ suc 𝐵))
185, 17sylan9r 508 . . . . . . . . . 10 ((Lim 𝐴𝑥𝐴) → (𝑥 = 𝐵 → ¬ 𝐴 ⊆ suc 𝐵))
1918con2d 134 . . . . . . . . 9 ((Lim 𝐴𝑥𝐴) → (𝐴 ⊆ suc 𝐵 → ¬ 𝑥 = 𝐵))
2019ex 412 . . . . . . . 8 (Lim 𝐴 → (𝑥𝐴 → (𝐴 ⊆ suc 𝐵 → ¬ 𝑥 = 𝐵)))
2120com23 86 . . . . . . 7 (Lim 𝐴 → (𝐴 ⊆ suc 𝐵 → (𝑥𝐴 → ¬ 𝑥 = 𝐵)))
2221imp31 417 . . . . . 6 (((Lim 𝐴𝐴 ⊆ suc 𝐵) ∧ 𝑥𝐴) → ¬ 𝑥 = 𝐵)
23 ssel2 3989 . . . . . . . . . 10 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → 𝑥 ∈ suc 𝐵)
24 vex 3481 . . . . . . . . . . 11 𝑥 ∈ V
2524elsuc 6455 . . . . . . . . . 10 (𝑥 ∈ suc 𝐵 ↔ (𝑥𝐵𝑥 = 𝐵))
2623, 25sylib 218 . . . . . . . . 9 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → (𝑥𝐵𝑥 = 𝐵))
2726ord 864 . . . . . . . 8 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → (¬ 𝑥𝐵𝑥 = 𝐵))
2827con1d 145 . . . . . . 7 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → (¬ 𝑥 = 𝐵𝑥𝐵))
2928adantll 714 . . . . . 6 (((Lim 𝐴𝐴 ⊆ suc 𝐵) ∧ 𝑥𝐴) → (¬ 𝑥 = 𝐵𝑥𝐵))
3022, 29mpd 15 . . . . 5 (((Lim 𝐴𝐴 ⊆ suc 𝐵) ∧ 𝑥𝐴) → 𝑥𝐵)
3130ex 412 . . . 4 ((Lim 𝐴𝐴 ⊆ suc 𝐵) → (𝑥𝐴𝑥𝐵))
3231ssrdv 4000 . . 3 ((Lim 𝐴𝐴 ⊆ suc 𝐵) → 𝐴𝐵)
3332ex 412 . 2 (Lim 𝐴 → (𝐴 ⊆ suc 𝐵𝐴𝐵))
343, 33impbid2 226 1 (Lim 𝐴 → (𝐴𝐵𝐴 ⊆ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1536  wcel 2105  wss 3962  Ord word 6384  Lim wlim 6386  suc csuc 6387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391
This theorem is referenced by:  cardlim  10009
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