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Theorem ordelordALT 41735
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6204 using the Axiom of Regularity indirectly through dford2 9168. 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 41735 is ordelordALTVD 42065 without virtual deductions and was automatically derived from ordelordALTVD 42065 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 6196 . . . 4 (Ord 𝐴 → Tr 𝐴)
21adantr 484 . . 3 ((Ord 𝐴𝐵𝐴) → Tr 𝐴)
3 dford2 9168 . . . . . 6 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
43simprbi 500 . . . . 5 (Ord 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
54adantr 484 . . . 4 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
6 3orcomb 1095 . . . . 5 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
762ralbii 3082 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
85, 7sylib 221 . . 3 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
9 simpr 488 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
10 tratrb 41734 . . 3 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
112, 8, 9, 10syl3anc 1372 . 2 ((Ord 𝐴𝐵𝐴) → Tr 𝐵)
12 trss 5155 . . . 4 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
132, 9, 12sylc 65 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
14 ssralv2 41729 . . . 4 ((𝐵𝐴𝐵𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
1514ex 416 . . 3 (𝐵𝐴 → (𝐵𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))))
1613, 13, 5, 15syl3c 66 . 2 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
17 dford2 9168 . 2 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
1811, 16, 17sylanbrc 586 1 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3o 1087  wcel 2114  wral 3054  wss 3853  Tr wtr 5146  Ord word 6181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491  ax-reg 9141
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-tp 4531  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-tr 5147  df-eprel 5444  df-po 5452  df-so 5453  df-fr 5493  df-we 5495  df-ord 6185
This theorem is referenced by: (None)
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