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Mirrors > Home > MPE Home > Th. List > Mathboxes > ellimits | Structured version Visualization version Unicode version |
Description: Membership in the class of all limit ordinals. (Contributed by Scott Fenton, 11-Apr-2012.) |
Ref | Expression |
---|---|
ellimits.1 |
Ref | Expression |
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ellimits |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-limits 31967 | . . 3 | |
2 | 1 | eleq2i 2693 | . 2 |
3 | eldif 3584 | . 2 | |
4 | 3anan32 1050 | . . 3 | |
5 | df-lim 5728 | . . 3 | |
6 | elin 3796 | . . . . 5 | |
7 | ellimits.1 | . . . . . . 7 | |
8 | 7 | elon 5732 | . . . . . 6 |
9 | 7 | elfix 32010 | . . . . . . 7 |
10 | 7 | brbigcup 32005 | . . . . . . 7 |
11 | eqcom 2629 | . . . . . . 7 | |
12 | 9, 10, 11 | 3bitri 286 | . . . . . 6 |
13 | 8, 12 | anbi12i 733 | . . . . 5 |
14 | 6, 13 | bitri 264 | . . . 4 |
15 | 7 | elsn 4192 | . . . . 5 |
16 | 15 | necon3bbii 2841 | . . . 4 |
17 | 14, 16 | anbi12i 733 | . . 3 |
18 | 4, 5, 17 | 3bitr4ri 293 | . 2 |
19 | 2, 3, 18 | 3bitri 286 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wne 2794 cvv 3200 cdif 3571 cin 3573 c0 3915 csn 4177 cuni 4436 class class class wbr 4653 word 5722 con0 5723 wlim 5724 cbigcup 31941 cfix 31942 climits 31943 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-symdif 3844 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ord 5726 df-on 5727 df-lim 5728 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fo 5894 df-fv 5896 df-1st 7168 df-2nd 7169 df-txp 31961 df-bigcup 31965 df-fix 31966 df-limits 31967 |
This theorem is referenced by: dfom5b 32019 dfrdg4 32058 |
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