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

Theorem ordunisssuc 5972
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 3747 . . . . 5 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
2 ordsssuc 5954 . . . . 5 ((𝑥 ∈ On ∧ Ord 𝐵) → (𝑥𝐵𝑥 ∈ suc 𝐵))
31, 2sylan 569 . . . 4 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ Ord 𝐵) → (𝑥𝐵𝑥 ∈ suc 𝐵))
43an32s 631 . . 3 (((𝐴 ⊆ On ∧ Ord 𝐵) ∧ 𝑥𝐴) → (𝑥𝐵𝑥 ∈ suc 𝐵))
54ralbidva 3134 . 2 ((𝐴 ⊆ On ∧ Ord 𝐵) → (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ suc 𝐵))
6 unissb 4606 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
7 dfss3 3741 . 2 (𝐴 ⊆ suc 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ suc 𝐵)
85, 6, 73bitr4g 303 1 ((𝐴 ⊆ On ∧ Ord 𝐵) → ( 𝐴𝐵𝐴 ⊆ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wcel 2145  wral 3061  wss 3723   cuni 4575  Ord word 5864  Oncon0 5865  suc csuc 5867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-ord 5868  df-on 5869  df-suc 5871
This theorem is referenced by:  ordsucuniel  7175  onsucuni  7179  isfinite2  8378  rankbnd2  8900
  Copyright terms: Public domain W3C validator