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Theorem ordelordALTVD 44865
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6408 using the Axiom of Regularity indirectly through dford2 9658. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that E Fr 𝐴 because this is inferred by the Axiom of Regularity. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ordelordALT 44535 is ordelordALTVD 44865 without virtual deductions and was automatically derived from ordelordALTVD 44865 using the tools program translate..without..overwriting.cmd and the Metamath program "MM-PA> MINIMIZE_WITH *" command.
1:: (   (Ord 𝐴𝐵𝐴)   ▶   (Ord 𝐴 𝐵𝐴)   )
2:1: (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐴   )
3:1: (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
4:2: (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐴   )
5:2: (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴 𝑦𝐴(𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
6:4,3: (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
7:6,6,5: (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐵 𝑦𝐵(𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
8:: ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
9:8: 𝑦((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
10:9: 𝑦𝐴((𝑥𝑦𝑥 = 𝑦 𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
11:10: (∀𝑦𝐴(𝑥𝑦𝑥 = 𝑦 𝑦𝑥) ↔ ∀𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦))
12:11: 𝑥(∀𝑦𝐴(𝑥𝑦𝑥 = 𝑦 𝑦𝑥) ↔ ∀𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦))
13:12: 𝑥𝐴(∀𝑦𝐴(𝑥𝑦 𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦))
14:13: (∀𝑥𝐴𝑦𝐴(𝑥𝑦 𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴(𝑥𝑦𝑦𝑥 𝑥 = 𝑦))
15:14,5: (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴 𝑦𝐴(𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
16:4,15,3: (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐵   )
17:16,7: (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐵   )
qed:17: ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
(Contributed by Alan Sare, 12-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ordelordALTVD ((Ord 𝐴𝐵𝐴) → Ord 𝐵)

Proof of Theorem ordelordALTVD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn1 44572 . . . . . 6 (   (Ord 𝐴𝐵𝐴)   ▶   (Ord 𝐴𝐵𝐴)   )
2 simpl 482 . . . . . 6 ((Ord 𝐴𝐵𝐴) → Ord 𝐴)
31, 2e1a 44625 . . . . 5 (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐴   )
4 ordtr 6400 . . . . 5 (Ord 𝐴 → Tr 𝐴)
53, 4e1a 44625 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐴   )
6 dford2 9658 . . . . . . 7 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
76simprbi 496 . . . . . 6 (Ord 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
83, 7e1a 44625 . . . . 5 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
9 3orcomb 1093 . . . . . . . . . . 11 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
109ax-gen 1792 . . . . . . . . . 10 𝑦((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
11 alral 3073 . . . . . . . . . 10 (∀𝑦((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ∀𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1210, 11e0a 44770 . . . . . . . . 9 𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
13 ralbi 3101 . . . . . . . . 9 (∀𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1412, 13e0a 44770 . . . . . . . 8 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
1514ax-gen 1792 . . . . . . 7 𝑥(∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
16 alral 3073 . . . . . . 7 (∀𝑥(∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ∀𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1715, 16e0a 44770 . . . . . 6 𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
18 ralbi 3101 . . . . . 6 (∀𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1917, 18e0a 44770 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
208, 19e1bi 44627 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
21 simpr 484 . . . . 5 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
221, 21e1a 44625 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
23 tratrb 44534 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
24233exp 1118 . . . 4 (Tr 𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝐵𝐴 → Tr 𝐵)))
255, 20, 22, 24e111 44672 . . 3 (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐵   )
26 trss 5276 . . . . 5 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
275, 22, 26e11 44686 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
28 ssralv2 44529 . . . . 5 ((𝐵𝐴𝐵𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
2928ex 412 . . . 4 (𝐵𝐴 → (𝐵𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))))
3027, 27, 8, 29e111 44672 . . 3 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
31 dford2 9658 . . . 4 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
3231simplbi2 500 . . 3 (Tr 𝐵 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → Ord 𝐵))
3325, 30, 32e11 44686 . 2 (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐵   )
3433in1 44569 1 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3o 1085  wal 1535   = wceq 1537  wcel 2106  wral 3059  wss 3963  Tr wtr 5265  Ord word 6385
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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-reg 9630
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-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  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-vd1 44568
This theorem is referenced by: (None)
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