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Theorem ordunisssuc 6492
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 3990 . . . . 5 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
2 ordsssuc 6475 . . . . 5 ((𝑥 ∈ On ∧ Ord 𝐵) → (𝑥𝐵𝑥 ∈ suc 𝐵))
31, 2sylan 580 . . . 4 (((𝐴 ⊆ On ∧ 𝑥𝐴) ∧ Ord 𝐵) → (𝑥𝐵𝑥 ∈ suc 𝐵))
43an32s 652 . . 3 (((𝐴 ⊆ On ∧ Ord 𝐵) ∧ 𝑥𝐴) → (𝑥𝐵𝑥 ∈ suc 𝐵))
54ralbidva 3174 . 2 ((𝐴 ⊆ On ∧ Ord 𝐵) → (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ suc 𝐵))
6 unissb 4944 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
7 dfss3 3984 . 2 (𝐴 ⊆ suc 𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ suc 𝐵)
85, 6, 73bitr4g 314 1 ((𝐴 ⊆ On ∧ Ord 𝐵) → ( 𝐴𝐵𝐴 ⊆ suc 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  wral 3059  wss 3963   cuni 4912  Ord word 6385  Oncon0 6386  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by:  ordsucuniel  7844  onsucuni  7848  isfinite2  9332  rankbnd2  9907  onintunirab  43216
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