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Theorem ordelordALTVD 43241
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6343 using the Axiom of Regularity indirectly through dford2 9564. 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 42911 is ordelordALTVD 43241 without virtual deductions and was automatically derived from ordelordALTVD 43241 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 42948 . . . . . 6 (   (Ord 𝐴𝐵𝐴)   ▶   (Ord 𝐴𝐵𝐴)   )
2 simpl 484 . . . . . 6 ((Ord 𝐴𝐵𝐴) → Ord 𝐴)
31, 2e1a 43001 . . . . 5 (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐴   )
4 ordtr 6335 . . . . 5 (Ord 𝐴 → Tr 𝐴)
53, 4e1a 43001 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐴   )
6 dford2 9564 . . . . . . 7 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
76simprbi 498 . . . . . 6 (Ord 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
83, 7e1a 43001 . . . . 5 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
9 3orcomb 1095 . . . . . . . . . . 11 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
109ax-gen 1798 . . . . . . . . . 10 𝑦((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
11 alral 3075 . . . . . . . . . 10 (∀𝑦((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ∀𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1210, 11e0a 43146 . . . . . . . . 9 𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
13 ralbi 3103 . . . . . . . . 9 (∀𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1412, 13e0a 43146 . . . . . . . 8 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
1514ax-gen 1798 . . . . . . 7 𝑥(∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
16 alral 3075 . . . . . . 7 (∀𝑥(∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ∀𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1715, 16e0a 43146 . . . . . 6 𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
18 ralbi 3103 . . . . . 6 (∀𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1917, 18e0a 43146 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
208, 19e1bi 43003 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
21 simpr 486 . . . . 5 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
221, 21e1a 43001 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
23 tratrb 42910 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
24233exp 1120 . . . 4 (Tr 𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝐵𝐴 → Tr 𝐵)))
255, 20, 22, 24e111 43048 . . 3 (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐵   )
26 trss 5237 . . . . 5 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
275, 22, 26e11 43062 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
28 ssralv2 42905 . . . . 5 ((𝐵𝐴𝐵𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
2928ex 414 . . . 4 (𝐵𝐴 → (𝐵𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))))
3027, 27, 8, 29e111 43048 . . 3 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
31 dford2 9564 . . . 4 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
3231simplbi2 502 . . 3 (Tr 𝐵 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → Ord 𝐵))
3325, 30, 32e11 43062 . 2 (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐵   )
3433in1 42945 1 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3o 1087  wal 1540   = wceq 1542  wcel 2107  wral 3061  wss 3914  Tr wtr 5226  Ord word 6320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676  ax-reg 9536
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-tr 5227  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-ord 6324  df-vd1 42944
This theorem is referenced by: (None)
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