MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordsucsssuc Structured version   Visualization version   GIF version

Theorem ordsucsssuc 7532
Description: The subclass relationship between two ordinal classes is inherited by their successors. (Contributed by NM, 4-Oct-2003.)
Assertion
Ref Expression
ordsucsssuc ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))

Proof of Theorem ordsucsssuc
StepHypRef Expression
1 ordsucelsuc 7531 . . . 4 (Ord 𝐴 → (𝐵𝐴 ↔ suc 𝐵 ∈ suc 𝐴))
21notbid 321 . . 3 (Ord 𝐴 → (¬ 𝐵𝐴 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
32adantr 484 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (¬ 𝐵𝐴 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
4 ordtri1 6211 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ ¬ 𝐵𝐴))
5 ordsuc 7523 . . 3 (Ord 𝐴 ↔ Ord suc 𝐴)
6 ordsuc 7523 . . 3 (Ord 𝐵 ↔ Ord suc 𝐵)
7 ordtri1 6211 . . 3 ((Ord suc 𝐴 ∧ Ord suc 𝐵) → (suc 𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
85, 6, 7syl2anb 600 . 2 ((Ord 𝐴 ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 ↔ ¬ suc 𝐵 ∈ suc 𝐴))
93, 4, 83bitr4d 314 1 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴𝐵 ↔ suc 𝐴 ⊆ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wcel 2115  wss 3919  Ord word 6177  suc csuc 6180
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-tr 5159  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-ord 6181  df-on 6182  df-suc 6184
This theorem is referenced by:  oawordri  8172  oeworde  8215  nnawordi  8243  bndrank  9267  rankmapu  9304  ackbij1b  9659  nosupbday  33262  onsuct0  33846  finxpsuclem  34759
  Copyright terms: Public domain W3C validator