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Theorem ordelordALTVD 28917
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4596 using the Axiom of Regularity indirectly through dford2 7568. 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 28560 is ordelordALTVD 28917 without virtual deductions and was automatically derived from ordelordALTVD 28917 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
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 idn1 28603 . . . . . 6
2 simpl 444 . . . . . 6
31, 2e1_ 28666 . . . . 5
4 ordtr 4588 . . . . 5
53, 4e1_ 28666 . . . 4
6 dford2 7568 . . . . . . 7
76simprbi 451 . . . . . 6
83, 7e1_ 28666 . . . . 5
9 3orcomb 946 . . . . . . . . . . 11
109ax-gen 1555 . . . . . . . . . 10
11 alral 2757 . . . . . . . . . 10
1210, 11e0_ 28822 . . . . . . . . 9
13 ralbi 2835 . . . . . . . . 9
1412, 13e0_ 28822 . . . . . . . 8
1514ax-gen 1555 . . . . . . 7
16 alral 2757 . . . . . . 7
1715, 16e0_ 28822 . . . . . 6
18 ralbi 2835 . . . . . 6
1917, 18e0_ 28822 . . . . 5
208, 19e1bi 28668 . . . 4
21 simpr 448 . . . . 5
221, 21e1_ 28666 . . . 4
23 tratrb 28558 . . . . 5
24233exp 1152 . . . 4
255, 20, 22, 24e111 28713 . . 3
26 trss 4304 . . . . 5
275, 22, 26e11 28727 . . . 4
28 ssralv2 28553 . . . . 5
2928ex 424 . . . 4
3027, 27, 8, 29e111 28713 . . 3
31 dford2 7568 . . . 4
3231simplbi2 609 . . 3
3325, 30, 32e11 28727 . 2
3433in1 28600 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3o 935  wal 1549   wceq 1652   wcel 1725  wral 2698   wss 3313   wtr 4295   word 4573 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396  ax-un 4694  ax-reg 7553 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-tp 3815  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-tr 4296  df-eprel 4487  df-po 4496  df-so 4497  df-fr 4534  df-we 4536  df-ord 4577  df-vd1 28599
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