Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ordelordALTVD Structured version   Visualization version   GIF version

Theorem ordelordALTVD 44367
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6387 using the Axiom of Regularity indirectly through dford2 9638. 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 44037 is ordelordALTVD 44367 without virtual deductions and was automatically derived from ordelordALTVD 44367 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 44074 . . . . . 6 (   (Ord 𝐴𝐵𝐴)   ▶   (Ord 𝐴𝐵𝐴)   )
2 simpl 481 . . . . . 6 ((Ord 𝐴𝐵𝐴) → Ord 𝐴)
31, 2e1a 44127 . . . . 5 (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐴   )
4 ordtr 6379 . . . . 5 (Ord 𝐴 → Tr 𝐴)
53, 4e1a 44127 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐴   )
6 dford2 9638 . . . . . . 7 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
76simprbi 495 . . . . . 6 (Ord 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
83, 7e1a 44127 . . . . 5 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
9 3orcomb 1091 . . . . . . . . . . 11 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
109ax-gen 1789 . . . . . . . . . 10 𝑦((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
11 alral 3065 . . . . . . . . . 10 (∀𝑦((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ∀𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1210, 11e0a 44272 . . . . . . . . 9 𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
13 ralbi 3093 . . . . . . . . 9 (∀𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1412, 13e0a 44272 . . . . . . . 8 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
1514ax-gen 1789 . . . . . . 7 𝑥(∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
16 alral 3065 . . . . . . 7 (∀𝑥(∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ∀𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1715, 16e0a 44272 . . . . . 6 𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
18 ralbi 3093 . . . . . 6 (∀𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1917, 18e0a 44272 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
208, 19e1bi 44129 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
21 simpr 483 . . . . 5 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
221, 21e1a 44127 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
23 tratrb 44036 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
24233exp 1116 . . . 4 (Tr 𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝐵𝐴 → Tr 𝐵)))
255, 20, 22, 24e111 44174 . . 3 (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐵   )
26 trss 5272 . . . . 5 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
275, 22, 26e11 44188 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
28 ssralv2 44031 . . . . 5 ((𝐵𝐴𝐵𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
2928ex 411 . . . 4 (𝐵𝐴 → (𝐵𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))))
3027, 27, 8, 29e111 44174 . . 3 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
31 dford2 9638 . . . 4 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
3231simplbi2 499 . . 3 (Tr 𝐵 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → Ord 𝐵))
3325, 30, 32e11 44188 . 2 (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐵   )
3433in1 44071 1 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3o 1083  wal 1531   = wceq 1533  wcel 2098  wral 3051  wss 3941  Tr wtr 5261  Ord word 6364
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424  ax-un 7735  ax-reg 9610
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-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-tr 5262  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-ord 6368  df-vd1 44070
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator