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Theorem ordunisssuc 6490
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 3978 . . . . 5 ((𝐴 ⊆ On ∧ 𝑥𝐴) → 𝑥 ∈ On)
2 ordsssuc 6473 . . . . 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 3176 . 2 ((𝐴 ⊆ On ∧ Ord 𝐵) → (∀𝑥𝐴 𝑥𝐵 ↔ ∀𝑥𝐴 𝑥 ∈ suc 𝐵))
6 unissb 4939 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
7 dfss3 3972 . 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 2108  wral 3061  wss 3951   cuni 4907  Ord word 6383  Oncon0 6384  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-tr 5260  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-ord 6387  df-on 6388  df-suc 6390
This theorem is referenced by:  ordsucuniel  7844  onsucuni  7848  isfinite2  9334  rankbnd2  9909  onintunirab  43239
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