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Theorem ordunisssuc 5747
 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 3563 . . . . 5 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
2 ordsssuc 5729 . . . . 5 ((𝑥 ∈ On ∧ Ord 𝐵) → (𝑥𝐵𝑥 ∈ suc 𝐵))
31, 2sylan 487 . . . 4 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ Ord 𝐵) → (𝑥𝐵𝑥 ∈ suc 𝐵))
43an32s 842 . . 3 (((𝐴 ⊆ On ∧ Ord 𝐵) ∧ 𝑥𝐴) → (𝑥𝐵𝑥 ∈ suc 𝐵))
54ralbidva 2968 . 2 ((𝐴 ⊆ On ∧ Ord 𝐵) → (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ suc 𝐵))
6 unissb 4405 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
7 dfss3 3558 . 2 (𝐴 ⊆ suc 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ suc 𝐵)
85, 6, 73bitr4g 302 1 ((𝐴 ⊆ On ∧ Ord 𝐵) → ( 𝐴𝐵𝐴 ⊆ suc 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ∪ cuni 4372  Ord word 5639  Oncon0 5640  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-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 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-sn 4126  df-pr 4128  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-suc 5646 This theorem is referenced by:  ordsucuniel  6916  onsucuni  6920  isfinite2  8103  rankbnd2  8615
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