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Theorem ordelordALT 42911
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 6343 using the Axiom of Regularity indirectly through dford2 9564. 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 42911 is ordelordALTVD 43241 without virtual deductions and was automatically derived from ordelordALTVD 43241 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 6335 . . . 4 (Ord 𝐴 → Tr 𝐴)
21adantr 482 . . 3 ((Ord 𝐴𝐵𝐴) → Tr 𝐴)
3 dford2 9564 . . . . . 6 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
43simprbi 498 . . . . 5 (Ord 𝐴 → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
54adantr 482 . . . 4 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
6 3orcomb 1095 . . . . 5 ((𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
762ralbii 3124 . . . 4 (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) ↔ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
85, 7sylib 217 . . 3 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦))
9 simpr 486 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
10 tratrb 42910 . . 3 ((Tr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑦𝑥𝑥 = 𝑦) ∧ 𝐵𝐴) → Tr 𝐵)
112, 8, 9, 10syl3anc 1372 . 2 ((Ord 𝐴𝐵𝐴) → Tr 𝐵)
12 trss 5237 . . . 4 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
132, 9, 12sylc 65 . . 3 ((Ord 𝐴𝐵𝐴) → 𝐵𝐴)
14 ssralv2 42905 . . . 4 ((𝐵𝐴𝐵𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
1514ex 414 . . 3 (𝐵𝐴 → (𝐵𝐴 → (∀𝑥𝐴𝑦𝐴 (𝑥𝑦𝑥 = 𝑦𝑦𝑥) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))))
1613, 13, 5, 15syl3c 66 . 2 ((Ord 𝐴𝐵𝐴) → ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥))
17 dford2 9564 . 2 (Ord 𝐵 ↔ (Tr 𝐵 ∧ ∀𝑥𝐵𝑦𝐵 (𝑥𝑦𝑥 = 𝑦𝑦𝑥)))
1811, 16, 17sylanbrc 584 1 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3o 1087  wcel 2107  wral 3061  wss 3914  Tr wtr 5226  Ord word 6320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676  ax-reg 9536
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-tr 5227  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-ord 6324
This theorem is referenced by: (None)
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