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Mirrors > Home > MPE Home > Th. List > ssceq | Structured version Visualization version Unicode version |
Description: The subcategory subset relation is antisymmetric. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
ssceq | cat cat |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . . . . 6 cat cat cat | |
2 | eqidd 2623 | . . . . . 6 cat cat | |
3 | 1, 2 | sscfn1 16477 | . . . . 5 cat cat |
4 | simpr 477 | . . . . . 6 cat cat cat | |
5 | eqidd 2623 | . . . . . 6 cat cat | |
6 | 4, 5 | sscfn1 16477 | . . . . 5 cat cat |
7 | 3, 6, 1 | ssc1 16481 | . . . 4 cat cat |
8 | 6, 3, 4 | ssc1 16481 | . . . 4 cat cat |
9 | 7, 8 | eqssd 3620 | . . 3 cat cat |
10 | 9 | sqxpeqd 5141 | . 2 cat cat |
11 | 3 | adantr 481 | . . . . 5 cat cat |
12 | 1 | adantr 481 | . . . . 5 cat cat cat |
13 | simprl 794 | . . . . 5 cat cat | |
14 | simprr 796 | . . . . 5 cat cat | |
15 | 11, 12, 13, 14 | ssc2 16482 | . . . 4 cat cat |
16 | 6 | adantr 481 | . . . . 5 cat cat |
17 | 4 | adantr 481 | . . . . 5 cat cat cat |
18 | 7 | adantr 481 | . . . . . 6 cat cat |
19 | 18, 13 | sseldd 3604 | . . . . 5 cat cat |
20 | 18, 14 | sseldd 3604 | . . . . 5 cat cat |
21 | 16, 17, 19, 20 | ssc2 16482 | . . . 4 cat cat |
22 | 15, 21 | eqssd 3620 | . . 3 cat cat |
23 | 22 | ralrimivva 2971 | . 2 cat cat |
24 | eqfnov 6766 | . . 3 | |
25 | 3, 6, 24 | syl2anc 693 | . 2 cat cat |
26 | 10, 23, 25 | mpbir2and 957 | 1 cat cat |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 wss 3574 class class class wbr 4653 cxp 5112 cdm 5114 wfn 5883 (class class class)co 6650 cat cssc 16467 |
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-rep 4771 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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 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-iun 4522 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-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-ixp 7909 df-ssc 16470 |
This theorem is referenced by: (None) |
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