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Theorem limuni3 4835
 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 4596 . . . . . . 7
21rspcv 3050 . . . . . 6
3 vex 2961 . . . . . . 7
4 limelon 4647 . . . . . . 7
53, 4mpan 653 . . . . . 6
62, 5syl6com 34 . . . . 5
76ssrdv 3356 . . . 4
8 ssorduni 4769 . . . 4
97, 8syl 16 . . 3
11 n0 3639 . . . 4
12 0ellim 4646 . . . . . . 7
13 elunii 4022 . . . . . . . 8
1413expcom 426 . . . . . . 7
1512, 14syl5 31 . . . . . 6
162, 15syld 43 . . . . 5
1716exlimiv 1645 . . . 4
1811, 17sylbi 189 . . 3
1918imp 420 . 2
20 eluni2 4021 . . . . 5
211rspccv 3051 . . . . . . 7
22 limsuc 4832 . . . . . . . . . . 11
2322anbi1d 687 . . . . . . . . . 10
24 elunii 4022 . . . . . . . . . 10
2523, 24syl6bi 221 . . . . . . . . 9
2625exp3a 427 . . . . . . . 8
2726com3r 76 . . . . . . 7
2821, 27sylcom 28 . . . . . 6
2928rexlimdv 2831 . . . . 5
3020, 29syl5bi 210 . . . 4
3130ralrimiv 2790 . . 3