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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 4067 | . . 3 | |
2 | simpr 109 | . . 3 | |
3 | f1oi 5405 | . . . . . . . . . 10 | |
4 | dff1o3 5373 | . . . . . . . . . 10 | |
5 | 3, 4 | mpbi 144 | . . . . . . . . 9 |
6 | 5 | simpli 110 | . . . . . . . 8 |
7 | fof 5345 | . . . . . . . 8 | |
8 | 6, 7 | ax-mp 5 | . . . . . . 7 |
9 | fss 5284 | . . . . . . 7 | |
10 | 8, 9 | mpan 420 | . . . . . 6 |
11 | funi 5155 | . . . . . . . 8 | |
12 | cnvi 4943 | . . . . . . . . 9 | |
13 | 12 | funeqi 5144 | . . . . . . . 8 |
14 | 11, 13 | mpbir 145 | . . . . . . 7 |
15 | funres11 5195 | . . . . . . 7 | |
16 | 14, 15 | ax-mp 5 | . . . . . 6 |
17 | 10, 16 | jctir 311 | . . . . 5 |
18 | df-f1 5128 | . . . . 5 | |
19 | 17, 18 | sylibr 133 | . . . 4 |
20 | 19 | adantr 274 | . . 3 |
21 | f1dom2g 6650 | . . 3 | |
22 | 1, 2, 20, 21 | syl3anc 1216 | . 2 |
23 | 22 | expcom 115 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wcel 1480 cvv 2686 wss 3071 class class class wbr 3929 cid 4210 ccnv 4538 cres 4541 wfun 5117 wf 5119 wf1 5120 wfo 5121 wf1o 5122 cdom 6633 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-dom 6636 |
This theorem is referenced by: cnvct 6703 ssct 6712 xpdom3m 6728 0domg 6731 mapdom1g 6741 phplem4dom 6756 nndomo 6758 phpm 6759 fict 6762 domfiexmid 6772 infnfi 6789 exmidfodomrlemr 7058 exmidfodomrlemrALT 7059 fihashss 10562 phicl2 11890 phibnd 11893 qnnen 11944 pw1dom2 13190 sbthom 13221 |
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