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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 4115 | . . 3 | |
2 | simpr 109 | . . 3 | |
3 | f1oi 5464 | . . . . . . . . . 10 | |
4 | dff1o3 5432 | . . . . . . . . . 10 | |
5 | 3, 4 | mpbi 144 | . . . . . . . . 9 |
6 | 5 | simpli 110 | . . . . . . . 8 |
7 | fof 5404 | . . . . . . . 8 | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 |
9 | fss 5343 | . . . . . . 7 | |
10 | 8, 9 | mpan 421 | . . . . . 6 |
11 | funi 5214 | . . . . . . . 8 | |
12 | cnvi 5002 | . . . . . . . . 9 | |
13 | 12 | funeqi 5203 | . . . . . . . 8 |
14 | 11, 13 | mpbir 145 | . . . . . . 7 |
15 | funres11 5254 | . . . . . . 7 | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 |
17 | 10, 16 | jctir 311 | . . . . 5 |
18 | df-f1 5187 | . . . . 5 | |
19 | 17, 18 | sylibr 133 | . . . 4 |
20 | 19 | adantr 274 | . . 3 |
21 | f1dom2g 6713 | . . 3 | |
22 | 1, 2, 20, 21 | syl3anc 1227 | . 2 |
23 | 22 | expcom 115 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 2135 cvv 2721 wss 3111 class class class wbr 3976 cid 4260 ccnv 4597 cres 4600 wfun 5176 wf 5178 wf1 5179 wfo 5180 wf1o 5181 cdom 6696 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-dom 6699 |
This theorem is referenced by: cnvct 6766 ssct 6775 xpdom3m 6791 0domg 6794 mapdom1g 6804 phplem4dom 6819 nndomo 6821 phpm 6822 fict 6825 domfiexmid 6835 infnfi 6852 exmidfodomrlemr 7149 exmidfodomrlemrALT 7150 pw1dom2 7174 fihashss 10718 phicl2 12125 phibnd 12128 qnnen 12307 sbthom 13746 |
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