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Theorem ordom 4846
 Description: Omega is ordinal. Theorem 7.32 of [TakeutiZaring] p. 43. (Contributed by NM, 18-Oct-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
ordom

Proof of Theorem ordom
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 4296 . . 3
2 onelon 4598 . . . . . . . 8
32expcom 425 . . . . . . 7
4 limord 4632 . . . . . . . . . . . 12
5 ordtr 4587 . . . . . . . . . . . 12
6 trel 4301 . . . . . . . . . . . 12
74, 5, 63syl 19 . . . . . . . . . . 11
87exp3a 426 . . . . . . . . . 10
98com12 29 . . . . . . . . 9
109a2d 24 . . . . . . . 8
1110alimdv 1631 . . . . . . 7
123, 11anim12d 547 . . . . . 6
13 elom 4840 . . . . . 6
14 elom 4840 . . . . . 6
1512, 13, 143imtr4g 262 . . . . 5
1615imp 419 . . . 4
1716ax-gen 1555 . . 3
181, 17mpgbir 1559 . 2
19 omsson 4841 . 2
20 ordon 4755 . 2