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Theorem ordssun 6040
Description: Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
ordssun ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶)))

Proof of Theorem ordssun
StepHypRef Expression
1 ordtri2or2 6037 . . 3 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶𝐶𝐵))
2 ssequn1 3981 . . . . . 6 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
3 sseq2 3823 . . . . . 6 ((𝐵𝐶) = 𝐶 → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐶))
42, 3sylbi 209 . . . . 5 (𝐵𝐶 → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐶))
5 olc 895 . . . . 5 (𝐴𝐶 → (𝐴𝐵𝐴𝐶))
64, 5syl6bi 245 . . . 4 (𝐵𝐶 → (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵𝐴𝐶)))
7 ssequn2 3984 . . . . . 6 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
8 sseq2 3823 . . . . . 6 ((𝐵𝐶) = 𝐵 → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐵))
97, 8sylbi 209 . . . . 5 (𝐶𝐵 → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐵))
10 orc 894 . . . . 5 (𝐴𝐵 → (𝐴𝐵𝐴𝐶))
119, 10syl6bi 245 . . . 4 (𝐶𝐵 → (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵𝐴𝐶)))
126, 11jaoi 884 . . 3 ((𝐵𝐶𝐶𝐵) → (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵𝐴𝐶)))
131, 12syl 17 . 2 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵𝐴𝐶)))
14 ssun 3990 . 2 ((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))
1513, 14impbid1 217 1 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wo 874   = wceq 1653  cun 3767  wss 3769  Ord word 5940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-tr 4946  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-ord 5944
This theorem is referenced by:  ordsucun  7259
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