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Theorem limsssuc 6942
 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 5719 . . 3 𝐵 ⊆ suc 𝐵
2 sstr2 3575 . . 3 (𝐴𝐵 → (𝐵 ⊆ suc 𝐵𝐴 ⊆ suc 𝐵))
31, 2mpi 20 . 2 (𝐴𝐵𝐴 ⊆ suc 𝐵)
4 eleq1 2676 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
54biimpcd 238 . . . . . . . . . . 11 (𝑥𝐴 → (𝑥 = 𝐵𝐵𝐴))
6 limsuc 6941 . . . . . . . . . . . . . 14 (Lim 𝐴 → (𝐵𝐴 ↔ suc 𝐵𝐴))
76biimpa 500 . . . . . . . . . . . . 13 ((Lim 𝐴𝐵𝐴) → suc 𝐵𝐴)
8 limord 5701 . . . . . . . . . . . . . . . 16 (Lim 𝐴 → Ord 𝐴)
98adantr 480 . . . . . . . . . . . . . . 15 ((Lim 𝐴𝐵𝐴) → Ord 𝐴)
10 ordelord 5662 . . . . . . . . . . . . . . . . 17 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
118, 10sylan 487 . . . . . . . . . . . . . . . 16 ((Lim 𝐴𝐵𝐴) → Ord 𝐵)
12 ordsuc 6906 . . . . . . . . . . . . . . . 16 (Ord 𝐵 ↔ Ord suc 𝐵)
1311, 12sylib 207 . . . . . . . . . . . . . . 15 ((Lim 𝐴𝐵𝐴) → Ord suc 𝐵)
14 ordtri1 5673 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵𝐴))
159, 13, 14syl2anc 691 . . . . . . . . . . . . . 14 ((Lim 𝐴𝐵𝐴) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵𝐴))
1615con2bid 343 . . . . . . . . . . . . 13 ((Lim 𝐴𝐵𝐴) → (suc 𝐵𝐴 ↔ ¬ 𝐴 ⊆ suc 𝐵))
177, 16mpbid 221 . . . . . . . . . . . 12 ((Lim 𝐴𝐵𝐴) → ¬ 𝐴 ⊆ suc 𝐵)
1817ex 449 . . . . . . . . . . 11 (Lim 𝐴 → (𝐵𝐴 → ¬ 𝐴 ⊆ suc 𝐵))
195, 18sylan9r 688 . . . . . . . . . 10 ((Lim 𝐴𝑥𝐴) → (𝑥 = 𝐵 → ¬ 𝐴 ⊆ suc 𝐵))
2019con2d 128 . . . . . . . . 9 ((Lim 𝐴𝑥𝐴) → (𝐴 ⊆ suc 𝐵 → ¬ 𝑥 = 𝐵))
2120ex 449 . . . . . . . 8 (Lim 𝐴 → (𝑥𝐴 → (𝐴 ⊆ suc 𝐵 → ¬ 𝑥 = 𝐵)))
2221com23 84 . . . . . . 7 (Lim 𝐴 → (𝐴 ⊆ suc 𝐵 → (𝑥𝐴 → ¬ 𝑥 = 𝐵)))
2322imp31 447 . . . . . 6 (((Lim 𝐴𝐴 ⊆ suc 𝐵) ∧ 𝑥𝐴) → ¬ 𝑥 = 𝐵)
24 ssel2 3563 . . . . . . . . . 10 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → 𝑥 ∈ suc 𝐵)
25 vex 3176 . . . . . . . . . . 11 𝑥 ∈ V
2625elsuc 5711 . . . . . . . . . 10 (𝑥 ∈ suc 𝐵 ↔ (𝑥𝐵𝑥 = 𝐵))
2724, 26sylib 207 . . . . . . . . 9 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → (𝑥𝐵𝑥 = 𝐵))
2827ord 391 . . . . . . . 8 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → (¬ 𝑥𝐵𝑥 = 𝐵))
2928con1d 138 . . . . . . 7 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → (¬ 𝑥 = 𝐵𝑥𝐵))
3029adantll 746 . . . . . 6 (((Lim 𝐴𝐴 ⊆ suc 𝐵) ∧ 𝑥𝐴) → (¬ 𝑥 = 𝐵𝑥𝐵))
3123, 30mpd 15 . . . . 5 (((Lim 𝐴𝐴 ⊆ suc 𝐵) ∧ 𝑥𝐴) → 𝑥𝐵)
3231ex 449 . . . 4 ((Lim 𝐴𝐴 ⊆ suc 𝐵) → (𝑥𝐴𝑥𝐵))
3332ssrdv 3574 . . 3 ((Lim 𝐴𝐴 ⊆ suc 𝐵) → 𝐴𝐵)
3433ex 449 . 2 (Lim 𝐴 → (𝐴 ⊆ suc 𝐵𝐴𝐵))
353, 34impbid2 215 1 (Lim 𝐴 → (𝐴𝐵𝐴 ⊆ suc 𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  Ord word 5639  Lim wlim 5641  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646 This theorem is referenced by:  cardlim  8681
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