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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfiota3 | Structured version Visualization version Unicode version |
Description: A definiton of iota using minimal quantifiers. (Contributed by Scott Fenton, 19-Feb-2013.) |
Ref | Expression |
---|---|
dfiota3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iota 5851 | . 2 | |
2 | abeq1 2733 | . . . . 5 | |
3 | exdistr 1919 | . . . . . 6 | |
4 | vex 3203 | . . . . . . . . 9 | |
5 | sneq 4187 | . . . . . . . . . 10 | |
6 | 5 | eqeq2d 2632 | . . . . . . . . 9 |
7 | 4, 6 | ceqsexv 3242 | . . . . . . . 8 |
8 | snex 4908 | . . . . . . . . . . 11 | |
9 | eqeq1 2626 | . . . . . . . . . . . . 13 | |
10 | eleq2 2690 | . . . . . . . . . . . . 13 | |
11 | 9, 10 | anbi12d 747 | . . . . . . . . . . . 12 |
12 | eqcom 2629 | . . . . . . . . . . . . 13 | |
13 | velsn 4193 | . . . . . . . . . . . . . 14 | |
14 | equcom 1945 | . . . . . . . . . . . . . 14 | |
15 | 13, 14 | bitri 264 | . . . . . . . . . . . . 13 |
16 | 12, 15 | anbi12ci 734 | . . . . . . . . . . . 12 |
17 | 11, 16 | syl6bb 276 | . . . . . . . . . . 11 |
18 | 8, 17 | ceqsexv 3242 | . . . . . . . . . 10 |
19 | an13 840 | . . . . . . . . . . 11 | |
20 | 19 | exbii 1774 | . . . . . . . . . 10 |
21 | 18, 20 | bitr3i 266 | . . . . . . . . 9 |
22 | 21 | exbii 1774 | . . . . . . . 8 |
23 | 7, 22 | bitr3i 266 | . . . . . . 7 |
24 | excom 2042 | . . . . . . 7 | |
25 | 23, 24 | bitri 264 | . . . . . 6 |
26 | eluniab 4447 | . . . . . 6 | |
27 | 3, 25, 26 | 3bitr4i 292 | . . . . 5 |
28 | 2, 27 | mpgbir 1726 | . . . 4 |
29 | df-sn 4178 | . . . . . . 7 | |
30 | dfsingles2 32028 | . . . . . . 7 | |
31 | 29, 30 | ineq12i 3812 | . . . . . 6 |
32 | inab 3895 | . . . . . . 7 | |
33 | 19.42v 1918 | . . . . . . . . 9 | |
34 | 33 | bicomi 214 | . . . . . . . 8 |
35 | 34 | abbii 2739 | . . . . . . 7 |
36 | 32, 35 | eqtri 2644 | . . . . . 6 |
37 | 31, 36 | eqtri 2644 | . . . . 5 |
38 | 37 | unieqi 4445 | . . . 4 |
39 | 28, 38 | eqtr4i 2647 | . . 3 |
40 | 39 | unieqi 4445 | . 2 |
41 | 1, 40 | eqtri 2644 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wex 1704 wcel 1990 cab 2608 cin 3573 csn 4177 cuni 4436 cio 5849 csingles 31946 |
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-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-eprel 5029 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 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-singleton 31969 df-singles 31970 |
This theorem is referenced by: dffv5 32031 |
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