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Theorem limsssuc 6919
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 5705 . . 3 𝐵 ⊆ suc 𝐵
2 sstr2 3574 . . 3 (𝐴𝐵 → (𝐵 ⊆ suc 𝐵𝐴 ⊆ suc 𝐵))
31, 2mpi 20 . 2 (𝐴𝐵𝐴 ⊆ suc 𝐵)
4 eleq1 2675 . . . . . . . . . . . 12 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
54biimpcd 237 . . . . . . . . . . 11 (𝑥𝐴 → (𝑥 = 𝐵𝐵𝐴))
6 limsuc 6918 . . . . . . . . . . . . . 14 (Lim 𝐴 → (𝐵𝐴 ↔ suc 𝐵𝐴))
76biimpa 499 . . . . . . . . . . . . 13 ((Lim 𝐴𝐵𝐴) → suc 𝐵𝐴)
8 limord 5687 . . . . . . . . . . . . . . . 16 (Lim 𝐴 → Ord 𝐴)
98adantr 479 . . . . . . . . . . . . . . 15 ((Lim 𝐴𝐵𝐴) → Ord 𝐴)
10 ordelord 5648 . . . . . . . . . . . . . . . . 17 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
118, 10sylan 486 . . . . . . . . . . . . . . . 16 ((Lim 𝐴𝐵𝐴) → Ord 𝐵)
12 ordsuc 6883 . . . . . . . . . . . . . . . 16 (Ord 𝐵 ↔ Ord suc 𝐵)
1311, 12sylib 206 . . . . . . . . . . . . . . 15 ((Lim 𝐴𝐵𝐴) → Ord suc 𝐵)
14 ordtri1 5659 . . . . . . . . . . . . . . 15 ((Ord 𝐴 ∧ Ord suc 𝐵) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵𝐴))
159, 13, 14syl2anc 690 . . . . . . . . . . . . . 14 ((Lim 𝐴𝐵𝐴) → (𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵𝐴))
1615con2bid 342 . . . . . . . . . . . . 13 ((Lim 𝐴𝐵𝐴) → (suc 𝐵𝐴 ↔ ¬ 𝐴 ⊆ suc 𝐵))
177, 16mpbid 220 . . . . . . . . . . . 12 ((Lim 𝐴𝐵𝐴) → ¬ 𝐴 ⊆ suc 𝐵)
1817ex 448 . . . . . . . . . . 11 (Lim 𝐴 → (𝐵𝐴 → ¬ 𝐴 ⊆ suc 𝐵))
195, 18sylan9r 687 . . . . . . . . . 10 ((Lim 𝐴𝑥𝐴) → (𝑥 = 𝐵 → ¬ 𝐴 ⊆ suc 𝐵))
2019con2d 127 . . . . . . . . 9 ((Lim 𝐴𝑥𝐴) → (𝐴 ⊆ suc 𝐵 → ¬ 𝑥 = 𝐵))
2120ex 448 . . . . . . . 8 (Lim 𝐴 → (𝑥𝐴 → (𝐴 ⊆ suc 𝐵 → ¬ 𝑥 = 𝐵)))
2221com23 83 . . . . . . 7 (Lim 𝐴 → (𝐴 ⊆ suc 𝐵 → (𝑥𝐴 → ¬ 𝑥 = 𝐵)))
2322imp31 446 . . . . . 6 (((Lim 𝐴𝐴 ⊆ suc 𝐵) ∧ 𝑥𝐴) → ¬ 𝑥 = 𝐵)
24 ssel2 3562 . . . . . . . . . 10 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → 𝑥 ∈ suc 𝐵)
25 vex 3175 . . . . . . . . . . 11 𝑥 ∈ V
2625elsuc 5697 . . . . . . . . . 10 (𝑥 ∈ suc 𝐵 ↔ (𝑥𝐵𝑥 = 𝐵))
2724, 26sylib 206 . . . . . . . . 9 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → (𝑥𝐵𝑥 = 𝐵))
2827ord 390 . . . . . . . 8 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → (¬ 𝑥𝐵𝑥 = 𝐵))
2928con1d 137 . . . . . . 7 ((𝐴 ⊆ suc 𝐵𝑥𝐴) → (¬ 𝑥 = 𝐵𝑥𝐵))
3029adantll 745 . . . . . 6 (((Lim 𝐴𝐴 ⊆ suc 𝐵) ∧ 𝑥𝐴) → (¬ 𝑥 = 𝐵𝑥𝐵))
3123, 30mpd 15 . . . . 5 (((Lim 𝐴𝐴 ⊆ suc 𝐵) ∧ 𝑥𝐴) → 𝑥𝐵)
3231ex 448 . . . 4 ((Lim 𝐴𝐴 ⊆ suc 𝐵) → (𝑥𝐴𝑥𝐵))
3332ssrdv 3573 . . 3 ((Lim 𝐴𝐴 ⊆ suc 𝐵) → 𝐴𝐵)
3433ex 448 . 2 (Lim 𝐴 → (𝐴 ⊆ suc 𝐵𝐴𝐵))
353, 34impbid2 214 1 (Lim 𝐴 → (𝐴𝐵𝐴 ⊆ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1976  wss 3539  Ord word 5625  Lim wlim 5627  suc csuc 5628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4939  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632
This theorem is referenced by:  cardlim  8658
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