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Mirrors > Home > MPE Home > Th. List > eqsstrri | Structured version Visualization version GIF version |
Description: Substitution of equality into a subclass relationship. (Contributed by NM, 19-Oct-1999.) |
Ref | Expression |
---|---|
eqsstr3.1 | ⊢ 𝐵 = 𝐴 |
eqsstr3.2 | ⊢ 𝐵 ⊆ 𝐶 |
Ref | Expression |
---|---|
eqsstrri | ⊢ 𝐴 ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsstr3.1 | . . 3 ⊢ 𝐵 = 𝐴 | |
2 | 1 | eqcomi 2745 | . 2 ⊢ 𝐴 = 𝐵 |
3 | eqsstr3.2 | . 2 ⊢ 𝐵 ⊆ 𝐶 | |
4 | 2, 3 | eqsstri 3960 | 1 ⊢ 𝐴 ⊆ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ⊆ wss 3892 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3439 df-in 3899 df-ss 3909 |
This theorem is referenced by: inss2 4169 dmv 5844 idssxp 5968 ofrfvalg 7573 ofval 7576 ofrval 7577 off 7583 ofres 7584 ofco 7588 dftpos4 8092 smores2 8216 dmttrcl 9523 rnttrcl 9524 onwf 9632 r0weon 9814 dju1dif 9974 unctb 10007 infmap2 10020 itunitc 10223 axcclem 10259 dfnn3 12033 cotr2 14733 ressbasss 16995 prdsle 17218 prdsless 17219 cntrss 18981 dprd2da 19690 opsrle 21293 indiscld 22287 leordtval2 22408 fiuncmp 22600 prdstopn 22824 ustneism 23420 itg1addlem4 24908 itg1addlem4OLD 24909 itg1addlem5 24910 tdeglem4OLD 25270 aannenlem3 25535 efifo 25748 konigsbergssiedgw 28659 pjoml4i 29994 5oai 30068 3oai 30075 bdopssadj 30488 xrge00 31340 xrge0mulc1cn 31936 esumdivc 32096 rpsqrtcn 32618 subfacp1lem5 33191 filnetlem3 34614 filnetlem4 34615 mblfinlem4 35861 itg2gt0cn 35876 psubspset 37800 psubclsetN 37992 dvrelog2 40114 dvrelog3 40115 relexpaddss 41364 corcltrcl 41385 relopabVD 42559 cncfiooicc 43484 amgmwlem 46564 |
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