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Mirrors > Home > MPE Home > Th. List > eqsstr3i | 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 |
---|---|
eqsstr3i | ⊢ 𝐴 ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqsstr3.1 | . . 3 ⊢ 𝐵 = 𝐴 | |
2 | 1 | eqcomi 2619 | . 2 ⊢ 𝐴 = 𝐵 |
3 | eqsstr3.2 | . 2 ⊢ 𝐵 ⊆ 𝐶 | |
4 | 2, 3 | eqsstri 3598 | 1 ⊢ 𝐴 ⊆ 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ⊆ wss 3540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-in 3547 df-ss 3554 |
This theorem is referenced by: inss2 3796 dmv 5262 ofrfval 6803 ofval 6804 ofrval 6805 off 6810 ofres 6811 ofco 6815 dftpos4 7258 smores2 7338 onwf 8576 r0weon 8718 cda1dif 8881 unctb 8910 infmap2 8923 itunitc 9126 axcclem 9162 dfnn3 10911 bcm1k 12964 bcpasc 12970 cotr2 13564 ressbasss 15759 strlemor1 15796 prdsle 15945 prdsless 15946 dprd2da 18264 opsrle 19296 indiscld 20705 leordtval2 20826 fiuncmp 21017 prdstopn 21241 ustneism 21837 itg1addlem4 23272 itg1addlem5 23273 tdeglem4 23624 aannenlem3 23889 efifo 24097 advlogexp 24201 pjoml4i 27830 5oai 27904 3oai 27911 bdopssadj 28324 xrge00 29017 xrge0mulc1cn 29315 esumdivc 29472 ballotlemodife 29886 subfacp1lem5 30420 filnetlem3 31545 filnetlem4 31546 mblfinlem4 32619 itg2gt0cn 32635 psubspset 34048 psubclsetN 34240 relexpaddss 37029 corcltrcl 37050 relopabVD 38159 cncfiooicc 38780 stoweidlem34 38927 konigsbergssiedgw 41419 amgmwlem 42357 |
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