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Theorem ordelordALTVD 37928
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 5648 using the Axiom of Regularity indirectly through dford2 8377. 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 37571 is ordelordALTVD 37928 without virtual deductions and was automatically derived from ordelordALTVD 37928 using the tools program translate..without..overwriting.cmd and Metamath's minimize 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 37614 . . . . . 6 (   (Ord 𝐴𝐵𝐴)   ▶   (Ord 𝐴𝐵𝐴)   )
2 simpl 471 . . . . . 6 ((Ord 𝐴𝐵𝐴) → Ord 𝐴)
31, 2e1a 37676 . . . . 5 (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐴   )
4 ordtr 5640 . . . . 5 (Ord 𝐴 → Tr 𝐴)
53, 4e1a 37676 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐴   )
6 dford2 8377 . . . . . . 7 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
76simprbi 478 . . . . . 6 (Ord 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
83, 7e1a 37676 . . . . 5 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
9 3orcomb 1040 . . . . . . . . . . 11 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
109ax-gen 1712 . . . . . . . . . 10 𝑦((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
11 alral 2911 . . . . . . . . . 10 (∀𝑦((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ∀𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1210, 11e0a 37823 . . . . . . . . 9 𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
13 ralbi 3049 . . . . . . . . 9 (∀𝑦𝐴 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1412, 13e0a 37823 . . . . . . . 8 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
1514ax-gen 1712 . . . . . . 7 𝑥(∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
16 alral 2911 . . . . . . 7 (∀𝑥(∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → ∀𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1715, 16e0a 37823 . . . . . 6 𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
18 ralbi 3049 . . . . . 6 (∀𝑥𝐴 (∀𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)))
1917, 18e0a 37823 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
208, 19e1bi 37678 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦)   )
21 simpr 475 . . . . 5 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
221, 21e1a 37676 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
23 tratrb 37570 . . . . 5 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
24233exp 1255 . . . 4 (Tr 𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) → (𝐵𝐴 → Tr 𝐵)))
255, 20, 22, 24e111 37723 . . 3 (   (Ord 𝐴𝐵𝐴)   ▶   Tr 𝐵   )
26 trss 4683 . . . . 5 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
275, 22, 26e11 37737 . . . 4 (   (Ord 𝐴𝐵𝐴)   ▶   𝐵𝐴   )
28 ssralv2 37561 . . . . 5 ((𝐵𝐴𝐵𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
2928ex 448 . . . 4 (𝐵𝐴 → (𝐵𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))))
3027, 27, 8, 29e111 37723 . . 3 (   (Ord 𝐴𝐵𝐴)   ▶   𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)   )
31 dford2 8377 . . . 4 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
3231simplbi2 652 . . 3 (Tr 𝐵 → (∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → Ord 𝐵))
3325, 30, 32e11 37737 . 2 (   (Ord 𝐴𝐵𝐴)   ▶   Ord 𝐵   )
3433in1 37611 1 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3o 1029  wal 1472   = wceq 1474  wcel 1976  wral 2895  wss 3539  Tr wtr 4674  Ord word 5625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824  ax-reg 8357
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-tr 4675  df-eprel 4939  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-ord 5629  df-vd1 37610
This theorem is referenced by: (None)
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