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Mirrors > Home > MPE Home > Th. List > ustn0 | Structured version Visualization version Unicode version |
Description: The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.) |
Ref | Expression |
---|---|
ustn0 | UnifOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 3919 | . . . . 5 | |
2 | 0ex 4790 | . . . . . 6 | |
3 | eleq2 2690 | . . . . . 6 | |
4 | 2, 3 | elab 3350 | . . . . 5 |
5 | 1, 4 | mtbir 313 | . . . 4 |
6 | vex 3203 | . . . . . . 7 | |
7 | selpw 4165 | . . . . . . . . . 10 | |
8 | 7 | abbii 2739 | . . . . . . . . 9 |
9 | abid2 2745 | . . . . . . . . . 10 | |
10 | 6, 6 | xpex 6962 | . . . . . . . . . . . 12 |
11 | 10 | pwex 4848 | . . . . . . . . . . 11 |
12 | 11 | pwex 4848 | . . . . . . . . . 10 |
13 | 9, 12 | eqeltri 2697 | . . . . . . . . 9 |
14 | 8, 13 | eqeltrri 2698 | . . . . . . . 8 |
15 | simp1 1061 | . . . . . . . . 9 | |
16 | 15 | ss2abi 3674 | . . . . . . . 8 |
17 | 14, 16 | ssexi 4803 | . . . . . . 7 |
18 | df-ust 22004 | . . . . . . . 8 UnifOn | |
19 | 18 | fvmpt2 6291 | . . . . . . 7 UnifOn |
20 | 6, 17, 19 | mp2an 708 | . . . . . 6 UnifOn |
21 | simp2 1062 | . . . . . . 7 | |
22 | 21 | ss2abi 3674 | . . . . . 6 |
23 | 20, 22 | eqsstri 3635 | . . . . 5 UnifOn |
24 | 23 | sseli 3599 | . . . 4 UnifOn |
25 | 5, 24 | mto 188 | . . 3 UnifOn |
26 | 25 | nex 1731 | . 2 UnifOn |
27 | 18 | funmpt2 5927 | . . . 4 UnifOn |
28 | elunirn 6509 | . . . 4 UnifOn UnifOn UnifOn UnifOn | |
29 | 27, 28 | ax-mp 5 | . . 3 UnifOn UnifOn UnifOn |
30 | ustfn 22005 | . . . . 5 UnifOn | |
31 | fndm 5990 | . . . . 5 UnifOn UnifOn | |
32 | 30, 31 | ax-mp 5 | . . . 4 UnifOn |
33 | 32 | rexeqi 3143 | . . 3 UnifOn UnifOn UnifOn |
34 | rexv 3220 | . . 3 UnifOn UnifOn | |
35 | 29, 33, 34 | 3bitri 286 | . 2 UnifOn UnifOn |
36 | 26, 35 | mtbir 313 | 1 UnifOn |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 w3a 1037 wceq 1483 wex 1704 wcel 1990 cab 2608 wral 2912 wrex 2913 cvv 3200 cin 3573 wss 3574 c0 3915 cpw 4158 cuni 4436 cid 5023 cxp 5112 ccnv 5113 cdm 5114 crn 5115 cres 5116 ccom 5118 wfun 5882 wfn 5883 cfv 5888 UnifOncust 22003 |
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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 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-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 df-ust 22004 |
This theorem is referenced by: (None) |
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