Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ssdomg | Unicode version |
Description: A set dominates its subsets. Theorem 16 of [Suppes] p. 94. (Contributed by NM, 19-Jun-1998.) (Revised by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
ssdomg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 3896 | . . 3 | |
2 | simpr 103 | . . 3 | |
3 | f1oi 5164 | . . . . . . . . . 10 | |
4 | dff1o3 5132 | . . . . . . . . . 10 | |
5 | 3, 4 | mpbi 133 | . . . . . . . . 9 |
6 | 5 | simpli 104 | . . . . . . . 8 |
7 | fof 5106 | . . . . . . . 8 | |
8 | 6, 7 | ax-mp 7 | . . . . . . 7 |
9 | fss 5054 | . . . . . . 7 | |
10 | 8, 9 | mpan 400 | . . . . . 6 |
11 | funi 4932 | . . . . . . . 8 | |
12 | cnvi 4728 | . . . . . . . . 9 | |
13 | 12 | funeqi 4922 | . . . . . . . 8 |
14 | 11, 13 | mpbir 134 | . . . . . . 7 |
15 | funres11 4971 | . . . . . . 7 | |
16 | 14, 15 | ax-mp 7 | . . . . . 6 |
17 | 10, 16 | jctir 296 | . . . . 5 |
18 | df-f1 4907 | . . . . 5 | |
19 | 17, 18 | sylibr 137 | . . . 4 |
20 | 19 | adantr 261 | . . 3 |
21 | f1dom2g 6236 | . . 3 | |
22 | 1, 2, 20, 21 | syl3anc 1135 | . 2 |
23 | 22 | expcom 109 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wcel 1393 cvv 2557 wss 2917 class class class wbr 3764 cid 4025 ccnv 4344 cres 4347 wfun 4896 wf 4898 wf1 4899 wfo 4900 wf1o 4901 cdom 6220 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-dom 6223 |
This theorem is referenced by: xpdom3m 6308 phplem4dom 6324 nndomo 6326 phpm 6327 |
Copyright terms: Public domain | W3C validator |