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Theorem ordssun 6473
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 6470 . . 3 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶𝐶𝐵))
2 ssequn1 4178 . . . . . 6 (𝐵𝐶 ↔ (𝐵𝐶) = 𝐶)
3 sseq2 4003 . . . . . 6 ((𝐵𝐶) = 𝐶 → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐶))
42, 3sylbi 216 . . . . 5 (𝐵𝐶 → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐶))
5 olc 866 . . . . 5 (𝐴𝐶 → (𝐴𝐵𝐴𝐶))
64, 5biimtrdi 252 . . . 4 (𝐵𝐶 → (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵𝐴𝐶)))
7 ssequn2 4181 . . . . . 6 (𝐶𝐵 ↔ (𝐵𝐶) = 𝐵)
8 sseq2 4003 . . . . . 6 ((𝐵𝐶) = 𝐵 → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐵))
97, 8sylbi 216 . . . . 5 (𝐶𝐵 → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐵))
10 orc 865 . . . . 5 (𝐴𝐵 → (𝐴𝐵𝐴𝐶))
119, 10biimtrdi 252 . . . 4 (𝐶𝐵 → (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵𝐴𝐶)))
126, 11jaoi 855 . . 3 ((𝐵𝐶𝐶𝐵) → (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵𝐴𝐶)))
131, 12syl 17 . 2 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵𝐴𝐶)))
14 ssun 4187 . 2 ((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))
1513, 14impbid1 224 1 ((Ord 𝐵 ∧ Ord 𝐶) → (𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵𝐴𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wo 845   = wceq 1533  cun 3942  wss 3944  Ord word 6370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6374
This theorem is referenced by:  ordsucun  7829
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