Theorem ordelordALT 38747

Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 5745 using the Axiom of Regularity indirectly through dford2 8517. 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. ordelordALT 38747 is ordelordALTVD 39103 without virtual deductions and was automatically derived from ordelordALTVD 39103 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ordelordALT

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

Step	Hyp	Ref	Expression
1		ordtr 5737	. . . 4 ⊢ (Ord 𝐴 → Tr 𝐴)
2	1	adantr 481	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Tr 𝐴)
3		dford2 8517	. . . . . 6 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
4	3	simprbi 480	. . . . 5 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
5	4	adantr 481	. . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
6		3orcomb 1048	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
7	6	2ralbii 2981	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
8	5, 7	sylib 208	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
9		simpr 477	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)
10		tratrb 38746	. . 3 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)
11	2, 8, 9, 10	syl3anc 1326	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)
12		trss 4761	. . . 4 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
13	2, 9, 12	sylc 65	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
14		ssralv2 38737	. . . 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)))
15	14	ex 450	. . 3 ⊢ (𝐵 ⊆ 𝐴 → (𝐵 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))))
16	13, 13, 5, 15	syl3c 66	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
17		dford2 8517	. 2 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
18	11, 16, 17	sylanbrc 698	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

Syntax hints:   → wi 4   ∧ wa 384   ∨ w3o 1036   ∈ wcel 1990  ∀wral 2912   ⊆ wss 3574  Tr wtr 4752  Ord word 5722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949  ax-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726

This theorem is referenced by: (None)