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Theorem ordelordALTVD 39852
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 5962 using the Axiom of Regularity indirectly through dford2 8766. 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 39512 is ordelordALTVD 39852 without virtual deductions and was automatically derived from ordelordALTVD 39852 using the tools program translate..without..overwriting.cmd and the Metamath program "MM-PA> MINIMIZEWITH *" 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 39549 . . . . . 6 (   (Ord 𝐴𝐵𝐴)   ▶   (Ord 𝐴𝐵𝐴)   )
2 simpl 475 . . . . . 6 ((Ord 𝐴𝐵𝐴) → Ord 𝐴)
31, 2e1a 39611 . . . . 5 (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐴   )
4 ordtr 5954 . . . . 5 (Ord 𝐴 → Tr 𝐴)
53, 4e1a 39611 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐴   )
6 dford2 8766 . . . . . . 7 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
76simprbi 491 . . . . . 6 (Ord 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
83, 7e1a 39611 . . . . 5 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
9 3orcomb 1115 . . . . . . . . . . 11 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
109ax-gen 1891 . . . . . . . . . 10 𝑦((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
11 alral 3108 . . . . . . . . . 10 (∀𝑦((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ∀𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1210, 11e0a 39757 . . . . . . . . 9 𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
13 ralbi 3248 . . . . . . . . 9 (∀𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1412, 13e0a 39757 . . . . . . . 8 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
1514ax-gen 1891 . . . . . . 7 𝑥(∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
16 alral 3108 . . . . . . 7 (∀𝑥(∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ∀𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1715, 16e0a 39757 . . . . . 6 𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
18 ralbi 3248 . . . . . 6 (∀𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1917, 18e0a 39757 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
208, 19e1bi 39613 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
21 simpr 478 . . . . 5 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
221, 21e1a 39611 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
23 tratrb 39511 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
24233exp 1149 . . . 4 (Tr 𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝐵𝐴 → Tr 𝐵)))
255, 20, 22, 24e111 39658 . . 3 (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐵   )
26 trss 4953 . . . . 5 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
275, 22, 26e11 39672 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
28 ssralv2 39506 . . . . 5 ((𝐵𝐴𝐵𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
2928ex 402 . . . 4 (𝐵𝐴 → (𝐵𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))))
3027, 27, 8, 29e111 39658 . . 3 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
31 dford2 8766 . . . 4 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
3231simplbi2 495 . . 3 (Tr 𝐵 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → Ord 𝐵))
3325, 30, 32e11 39672 . 2 (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐵   )
3433in1 39546 1 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3o 1107  wal 1651   = wceq 1653  wcel 2157  wral 3088  wss 3768  Tr wtr 4944  Ord word 5939
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-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096  ax-un 7182  ax-reg 8738
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 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-sbc 3633  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4628  df-br 4843  df-opab 4905  df-tr 4945  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-ord 5943  df-vd1 39545
This theorem is referenced by: (None)
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