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Theorem ordelordALT 39064
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 5783 using the Axiom of Regularity indirectly through dford2 8555. 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 39064 is ordelordALTVD 39417 without virtual deductions and was automatically derived from ordelordALTVD 39417 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 𝐴𝐵𝐴) → Ord 𝐵)

Proof of Theorem ordelordALT
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtr 5775 . . . 4 (Ord 𝐴 → Tr 𝐴)
21adantr 480 . . 3 ((Ord 𝐴𝐵𝐴) → Tr 𝐴)
3 dford2 8555 . . . . . 6 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
43simprbi 479 . . . . 5 (Ord 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
54adantr 480 . . . 4 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
6 3orcomb 1065 . . . . 5 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
762ralbii 3010 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
85, 7sylib 208 . . 3 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
9 simpr 476 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
10 tratrb 39063 . . 3 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
112, 8, 9, 10syl3anc 1366 . 2 ((Ord 𝐴𝐵𝐴) → Tr 𝐵)
12 trss 4794 . . . 4 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
132, 9, 12sylc 65 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
14 ssralv2 39054 . . . 4 ((𝐵𝐴𝐵𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
1514ex 449 . . 3 (𝐵𝐴 → (𝐵𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))))
1613, 13, 5, 15syl3c 66 . 2 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
17 dford2 8555 . 2 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
1811, 16, 17sylanbrc 699 1 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3o 1053  wcel 2030  wral 2941  wss 3607  Tr wtr 4785  Ord word 5760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764
This theorem is referenced by: (None)
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