Theorem ordelordALTVD 45049

Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6337 using the Axiom of Regularity indirectly through dford2 9527. 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 44720 is ordelordALTVD 45049 without virtual deductions and was automatically derived from ordelordALTVD 45049 using the tools program translate..without..overwriting.cmd and the Metamath program "MM-PA> MINIMIZE_WITH *" command.

1::	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   )
2:1:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Ord 𝐴   )
3:1:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   𝐵 ∈ 𝐴   )
4:2:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Tr 𝐴   )
5:2:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)   )
6:4,3:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   𝐵 ⊆ 𝐴   )
7:6,6,5:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)   )
8::	⊢ ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
9:8:	⊢ ∀𝑦((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
10:9:	⊢ ∀𝑦 ∈ 𝐴((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
11:10:	⊢ (∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
12:11:	⊢ ∀𝑥(∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
13:12:	⊢ ∀𝑥 ∈ 𝐴(∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
14:13:	⊢ (∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐴∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
15:14,5:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴(𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)   )
16:4,15,3:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Tr 𝐵   )
17:16,7:	⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Ord 𝐵   )
qed:17:	⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

(Contributed by Alan Sare, 12-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ordelordALTVD	⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

Proof of Theorem ordelordALTVD

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		idn1 44757	. . . . . 6 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   )
2		simpl 482	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐴)
3	1, 2	e1a 44810	. . . . 5 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Ord 𝐴   )
4		ordtr 6329	. . . . 5 ⊢ (Ord 𝐴 → Tr 𝐴)
5	3, 4	e1a 44810	. . . 4 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Tr 𝐴   )
6		dford2 9527	. . . . . . 7 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
7	6	simprbi 496	. . . . . 6 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
8	3, 7	e1a 44810	. . . . 5 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)   )
9		3orcomb 1093	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
10	9	ax-gen 1796	. . . . . . . . . 10 ⊢ ∀𝑦((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
11		alral 3063	. . . . . . . . . 10 ⊢ (∀𝑦((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)) → ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)))
12	10, 11	e0a 44954	. . . . . . . . 9 ⊢ ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
13		ralbi 3089	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)) → (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)))
14	12, 13	e0a 44954	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
15	14	ax-gen 1796	. . . . . . 7 ⊢ ∀𝑥(∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
16		alral 3063	. . . . . . 7 ⊢ (∀𝑥(∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)) → ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)))
17	15, 16	e0a 44954	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
18		ralbi 3089	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)))
19	17, 18	e0a 44954	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
20	8, 19	e1bi 44812	. . . 4 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)   )
21		simpr 484	. . . . 5 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)
22	1, 21	e1a 44810	. . . 4 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   𝐵 ∈ 𝐴   )
23		tratrb 44719	. . . . 5 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)
24	23	3exp 1119	. . . 4 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → (𝐵 ∈ 𝐴 → Tr 𝐵)))
25	5, 20, 22, 24	e111 44857	. . 3 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Tr 𝐵   )
26		trss 5213	. . . . 5 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
27	5, 22, 26	e11 44871	. . . 4 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   𝐵 ⊆ 𝐴   )
28		ssralv2 44714	. . . . 5 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
29	28	ex 412	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))))
30	27, 27, 8, 29	e111 44857	. . 3 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)   )
31		dford2 9527	. . . 4 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
32	31	simplbi2 500	. . 3 ⊢ (Tr 𝐵 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → Ord 𝐵))
33	25, 30, 32	e11 44871	. 2 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Ord 𝐵   )
34	33	in1 44754	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085  ∀wal 1539   = wceq 1541   ∈ wcel 2113  ∀wral 3049   ⊆ wss 3899  Tr wtr 5203  Ord word 6314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678  ax-reg 9495

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-tr 5204  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-ord 6318  df-vd1 44753

This theorem is referenced by: (None)