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Theorem oneqmini 4624
 Description: A way to show that an ordinal number equals the minimum of a collection of ordinal numbers: it must be in the collection, and it must not be larger than any member of the collection. (Contributed by NM, 14-Nov-2003.)
Assertion
Ref Expression
oneqmini
Distinct variable groups:   ,   ,

Proof of Theorem oneqmini
StepHypRef Expression
1 ssint 4058 . . . . . 6
2 ssel 3334 . . . . . . . . . . . 12
3 ssel 3334 . . . . . . . . . . . 12
42, 3anim12d 547 . . . . . . . . . . 11
5 ontri1 4607 . . . . . . . . . . 11
64, 5syl6 31 . . . . . . . . . 10
76expdimp 427 . . . . . . . . 9
87pm5.74d 239 . . . . . . . 8
9 con2b 325 . . . . . . . 8
108, 9syl6bb 253 . . . . . . 7
1110ralbidv2 2719 . . . . . 6
121, 11syl5bb 249 . . . . 5
1312biimprd 215 . . . 4
1413expimpd 587 . . 3
15 intss1 4057 . . . . 5
1615a1i 11 . . . 4