Theorem ordelordALTVD 44219

Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6385 using the Axiom of Regularity indirectly through dford2 9629. 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 43889 is ordelordALTVD 44219 without virtual deductions and was automatically derived from ordelordALTVD 44219 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 43926	. . . . . 6 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   )
2		simpl 482	. . . . . 6 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐴)
3	1, 2	e1a 43979	. . . . 5 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Ord 𝐴   )
4		ordtr 6377	. . . . 5 ⊢ (Ord 𝐴 → Tr 𝐴)
5	3, 4	e1a 43979	. . . 4 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Tr 𝐴   )
6		dford2 9629	. . . . . . 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 43979	. . . . 5 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)   )
9		3orcomb 1092	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
10	9	ax-gen 1790	. . . . . . . . . 10 ⊢ ∀𝑦((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
11		alral 3070	. . . . . . . . . 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 44124	. . . . . . . . 9 ⊢ ∀𝑦 ∈ 𝐴 ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
13		ralbi 3098	. . . . . . . . 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 44124	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
15	14	ax-gen 1790	. . . . . . 7 ⊢ ∀𝑥(∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
16		alral 3070	. . . . . . 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 44124	. . . . . 6 ⊢ ∀𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
18		ralbi 3098	. . . . . 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 44124	. . . . 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 43981	. . . 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 43979	. . . 4 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   𝐵 ∈ 𝐴   )
23		tratrb 43888	. . . . 5 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)
24	23	3exp 1117	. . . 4 ⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → (𝐵 ∈ 𝐴 → Tr 𝐵)))
25	5, 20, 22, 24	e111 44026	. . 3 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Tr 𝐵   )
26		trss 5270	. . . . 5 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
27	5, 22, 26	e11 44040	. . . 4 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   𝐵 ⊆ 𝐴   )
28		ssralv2 43883	. . . . 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 44026	. . 3 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)   )
31		dford2 9629	. . . 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 44040	. 2 ⊢ (   (Ord 𝐴 ∧ 𝐵 ∈ 𝐴)   ▶   Ord 𝐵   )
34	33	in1 43923	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ w3o 1084  ∀wal 1532   = wceq 1534   ∈ wcel 2099  ∀wral 3056   ⊆ wss 3944  Tr wtr 5259  Ord word 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7732  ax-reg 9601

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-vd1 43922

This theorem is referenced by: (None)