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Theorem limuni3 6684
 Description: The union of a nonempty class of limit ordinals is a limit ordinal. (Contributed by NM, 1-Feb-2005.)
Assertion
Ref Expression
limuni3
Distinct variable group:   ,

Proof of Theorem limuni3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limeq 5438 . . . . . . 7
21rspcv 3148 . . . . . 6
3 vex 3050 . . . . . . 7
4 limelon 5489 . . . . . . 7
53, 4mpan 677 . . . . . 6
62, 5syl6com 36 . . . . 5
76ssrdv 3440 . . . 4
8 ssorduni 6617 . . . 4
97, 8syl 17 . . 3
11 n0 3743 . . . 4
12 0ellim 5488 . . . . . . 7
13 elunii 4206 . . . . . . . 8
1413expcom 437 . . . . . . 7
1512, 14syl5 33 . . . . . 6
162, 15syld 45 . . . . 5
1716exlimiv 1778 . . . 4
1811, 17sylbi 199 . . 3
1918imp 431 . 2
20 eluni2 4205 . . . . 5
211rspccv 3149 . . . . . . 7
22 limsuc 6681 . . . . . . . . . . 11
2322anbi1d 712 . . . . . . . . . 10
24 elunii 4206 . . . . . . . . . 10
2523, 24syl6bi 232 . . . . . . . . 9
2625expd 438 . . . . . . . 8
2726com3r 82 . . . . . . 7
2821, 27sylcom 30 . . . . . 6
2928rexlimdv 2879 . . . . 5
3020, 29syl5bi 221 . . . 4
3130ralrimiv 2802 . . 3