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

Theorem ordunisssuc 6010
Description: A subclass relationship for union and successor of ordinal classes. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
ordunisssuc ((𝐴 ⊆ On ∧ Ord 𝐵) → ( 𝐴𝐵𝐴 ⊆ suc 𝐵))

Proof of Theorem ordunisssuc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssel2 3756 . . . . 5 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
2 ordsssuc 5994 . . . . 5 ((𝑥 ∈ On ∧ Ord 𝐵) → (𝑥𝐵𝑥 ∈ suc 𝐵))
31, 2sylan 575 . . . 4 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ Ord 𝐵) → (𝑥𝐵𝑥 ∈ suc 𝐵))
43an32s 642 . . 3 (((𝐴 ⊆ On ∧ Ord 𝐵) ∧ 𝑥𝐴) → (𝑥𝐵𝑥 ∈ suc 𝐵))
54ralbidva 3132 . 2 ((𝐴 ⊆ On ∧ Ord 𝐵) → (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ suc 𝐵))
6 unissb 4627 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
7 dfss3 3750 . 2 (𝐴 ⊆ suc 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ suc 𝐵)
85, 6, 73bitr4g 305 1 ((𝐴 ⊆ On ∧ Ord 𝐵) → ( 𝐴𝐵𝐴 ⊆ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wcel 2155  wral 3055  wss 3732   cuni 4594  Ord word 5907  Oncon0 5908  suc csuc 5910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-tr 4912  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-ord 5911  df-on 5912  df-suc 5914
This theorem is referenced by:  ordsucuniel  7222  onsucuni  7226  isfinite2  8425  rankbnd2  8947
  Copyright terms: Public domain W3C validator