Theorem ordelordALT 39415

Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 5930 using the Axiom of Regularity indirectly through dford2 8732. 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 39415 is ordelordALTVD 39755 without virtual deductions and was automatically derived from ordelordALTVD 39755 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 5922	. . . 4 ⊢ (Ord 𝐴 → Tr 𝐴)
2	1	adantr 472	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Tr 𝐴)
3		dford2 8732	. . . . . 6 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
4	3	simprbi 490	. . . . 5 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
5	4	adantr 472	. . . 4 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
6		3orcomb 1114	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
7	6	2ralbii 3128	. . . 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 209	. . 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 39414	. . 3 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)
11	2, 8, 9, 10	syl3anc 1490	. 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)
12		trss 4920	. . . 4 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
13	2, 9, 12	sylc 65	. . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → 𝐵 ⊆ 𝐴)
14		ssralv2 39409	. . . 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 401	. . 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 8732	. 2 ⊢ (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)))
18	11, 16, 17	sylanbrc 578	1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ 𝐴) → Ord 𝐵)

Syntax hints:   → wi 4   ∧ wa 384   ∨ w3o 1106   ∈ wcel 2155  ∀wral 3055   ⊆ wss 3732  Tr wtr 4911  Ord word 5907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147  ax-reg 8704

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-tr 4912  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-ord 5911

This theorem is referenced by: (None)