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Theorem ordelordALTVD 27333
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4307 using the Axiom of Regularity indirectly through dford2 7205. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that because this is inferred by the Axiom of Regularity. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. ordelordALT 26994 is ordelordALTVD 27333 without virtual deductions and was automatically derived from ordelordALTVD 27333 using the tools program translate..without..overwriting.cmd and Metamath's minimize command.
 1:: 2:1: 3:1: 4:2: 5:2: 6:4,3: 7:6,6,5: 8:: 9:8: 10:9: 11:10: 12:11: 13:12: 14:13: 15:14,5: 16:4,15,3: 17:16,7: qed:17:
(Contributed by Alan Sare, 12-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ordelordALTVD

Proof of Theorem ordelordALTVD
StepHypRef Expression
1 idn1 27035 . . . . . 6
2 simpl 445 . . . . . 6
31, 2e1_ 27089 . . . . 5
4 ordtr 4299 . . . . 5
53, 4e1_ 27089 . . . 4
6 dford2 7205 . . . . . . 7
76simprbi 452 . . . . . 6
83, 7e1_ 27089 . . . . 5
9 3orcomb 949 . . . . . . . . . . 11
109ax-gen 1536 . . . . . . . . . 10
11 alral 2563 . . . . . . . . . 10
1210, 11e0_ 27237 . . . . . . . . 9
13 ralbi 2641 . . . . . . . . 9
1412, 13e0_ 27237 . . . . . . . 8
1514ax-gen 1536 . . . . . . 7
16 alral 2563 . . . . . . 7
1715, 16e0_ 27237 . . . . . 6
18 ralbi 2641 . . . . . 6
1917, 18e0_ 27237 . . . . 5
208, 19e1bi 27091 . . . 4
21 simpr 449 . . . . 5
221, 21e1_ 27089 . . . 4
23 tratrb 26992 . . . . 5
24233exp 1155 . . . 4
255, 20, 22, 24e111 27136 . . 3
26 trss 4019 . . . . 5
275, 22, 26e11 27150 . . . 4
28 ssralv2 26987 . . . . 5
2928ex 425 . . . 4
3027, 27, 8, 29e111 27136 . . 3
31 dford2 7205 . . . 4
3231simplbi2 611 . . 3
3325, 30, 32e11 27150 . 2
3433in1 27032 1
 Colors of variables: wff set class Syntax hints:   wi 6   wb 178   wa 360   w3o 938  wal 1532   wceq 1619   wcel 1621  wral 2509   wss 3078   wtr 4010   word 4284 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403  ax-reg 7190 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-vd1 27031
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