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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfon3 | Structured version Visualization version Unicode version |
Description: A quantifier-free definition of . (Contributed by Scott Fenton, 5-Apr-2012.) |
Ref | Expression |
---|---|
dfon3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfon2 31697 | . 2 | |
2 | abeq1 2733 | . . 3 | |
3 | vex 3203 | . . . . . . 7 | |
4 | 3 | elrn 5366 | . . . . . 6 |
5 | brin 4704 | . . . . . . . . . . 11 | |
6 | 3 | brsset 31996 | . . . . . . . . . . . 12 |
7 | brxp 5147 | . . . . . . . . . . . . . 14 | |
8 | 3, 7 | mpbiran2 954 | . . . . . . . . . . . . 13 |
9 | vex 3203 | . . . . . . . . . . . . . 14 | |
10 | 9 | eltrans 31998 | . . . . . . . . . . . . 13 |
11 | 8, 10 | bitri 264 | . . . . . . . . . . . 12 |
12 | 6, 11 | anbi12i 733 | . . . . . . . . . . 11 |
13 | 5, 12 | bitri 264 | . . . . . . . . . 10 |
14 | ioran 511 | . . . . . . . . . . 11 | |
15 | brun 4703 | . . . . . . . . . . . 12 | |
16 | 3 | ideq 5274 | . . . . . . . . . . . . 13 |
17 | epel 5032 | . . . . . . . . . . . . 13 | |
18 | 16, 17 | orbi12i 543 | . . . . . . . . . . . 12 |
19 | 15, 18 | bitri 264 | . . . . . . . . . . 11 |
20 | 14, 19 | xchnxbir 323 | . . . . . . . . . 10 |
21 | 13, 20 | anbi12i 733 | . . . . . . . . 9 |
22 | brdif 4705 | . . . . . . . . 9 | |
23 | dfpss2 3692 | . . . . . . . . . . . . 13 | |
24 | 23 | anbi1i 731 | . . . . . . . . . . . 12 |
25 | an32 839 | . . . . . . . . . . . 12 | |
26 | 24, 25 | bitri 264 | . . . . . . . . . . 11 |
27 | 26 | anbi1i 731 | . . . . . . . . . 10 |
28 | anass 681 | . . . . . . . . . 10 | |
29 | 27, 28 | bitri 264 | . . . . . . . . 9 |
30 | 21, 22, 29 | 3bitr4i 292 | . . . . . . . 8 |
31 | 30 | exbii 1774 | . . . . . . 7 |
32 | exanali 1786 | . . . . . . 7 | |
33 | 31, 32 | bitri 264 | . . . . . 6 |
34 | 4, 33 | bitri 264 | . . . . 5 |
35 | 34 | con2bii 347 | . . . 4 |
36 | eldif 3584 | . . . . 5 | |
37 | 3, 36 | mpbiran 953 | . . . 4 |
38 | 35, 37 | bitr4i 267 | . . 3 |
39 | 2, 38 | mpgbir 1726 | . 2 |
40 | 1, 39 | eqtri 2644 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 wal 1481 wceq 1483 wex 1704 wcel 1990 cab 2608 cvv 3200 cdif 3571 cun 3572 cin 3573 wss 3574 wpss 3575 class class class wbr 4653 wtr 4752 cid 5023 cep 5028 cxp 5112 crn 5115 con0 5723 csset 31939 ctrans 31940 |
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-3or 1038 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 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-suc 5729 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-sset 31963 df-trans 31964 |
This theorem is referenced by: dfon4 32000 |
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