MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordelinelOLD Structured version   Visualization version   GIF version

Theorem ordelinelOLD 5864
Description: Obsolete proof of ordelinel 5863 as of 24-Sep-2021. (Contributed by David Moews, 1-May-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
ordelinelOLD ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))

Proof of Theorem ordelinelOLD
StepHypRef Expression
1 ordtri2or3 5862 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
213adant3 1101 . . 3 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)))
3 eleq1 2718 . . . . 5 (𝐴 = (𝐴𝐵) → (𝐴𝐶 ↔ (𝐴𝐵) ∈ 𝐶))
4 orc 399 . . . . 5 (𝐴𝐶 → (𝐴𝐶𝐵𝐶))
53, 4syl6bir 244 . . . 4 (𝐴 = (𝐴𝐵) → ((𝐴𝐵) ∈ 𝐶 → (𝐴𝐶𝐵𝐶)))
6 eleq1 2718 . . . . 5 (𝐵 = (𝐴𝐵) → (𝐵𝐶 ↔ (𝐴𝐵) ∈ 𝐶))
7 olc 398 . . . . 5 (𝐵𝐶 → (𝐴𝐶𝐵𝐶))
86, 7syl6bir 244 . . . 4 (𝐵 = (𝐴𝐵) → ((𝐴𝐵) ∈ 𝐶 → (𝐴𝐶𝐵𝐶)))
95, 8jaoi 393 . . 3 ((𝐴 = (𝐴𝐵) ∨ 𝐵 = (𝐴𝐵)) → ((𝐴𝐵) ∈ 𝐶 → (𝐴𝐶𝐵𝐶)))
102, 9syl 17 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 → (𝐴𝐶𝐵𝐶)))
11 inss1 3866 . . . 4 (𝐴𝐵) ⊆ 𝐴
12 ordin 5791 . . . . 5 ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴𝐵))
13 ordtr2 5806 . . . . 5 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (((𝐴𝐵) ⊆ 𝐴𝐴𝐶) → (𝐴𝐵) ∈ 𝐶))
1412, 13stoic3 1741 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (((𝐴𝐵) ⊆ 𝐴𝐴𝐶) → (𝐴𝐵) ∈ 𝐶))
1511, 14mpani 712 . . 3 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐴𝐶 → (𝐴𝐵) ∈ 𝐶))
16 inss2 3867 . . . 4 (𝐴𝐵) ⊆ 𝐵
17 ordtr2 5806 . . . . 5 ((Ord (𝐴𝐵) ∧ Ord 𝐶) → (((𝐴𝐵) ⊆ 𝐵𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
1812, 17stoic3 1741 . . . 4 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (((𝐴𝐵) ⊆ 𝐵𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
1916, 18mpani 712 . . 3 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → (𝐵𝐶 → (𝐴𝐵) ∈ 𝐶))
2015, 19jaod 394 . 2 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶))
2110, 20impbid 202 1 ((Ord 𝐴 ∧ Ord 𝐵 ∧ Ord 𝐶) → ((𝐴𝐵) ∈ 𝐶 ↔ (𝐴𝐶𝐵𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  cin 3606  wss 3607  Ord word 5760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator