Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eqimss2i | Structured version Visualization version GIF version |
Description: Infer subclass relationship from equality. (Contributed by NM, 7-Jan-2007.) |
Ref | Expression |
---|---|
eqimssi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
eqimss2i | ⊢ 𝐵 ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3624 | . 2 ⊢ 𝐵 ⊆ 𝐵 | |
2 | eqimssi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
3 | 1, 2 | sseqtr4i 3638 | 1 ⊢ 𝐵 ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ⊆ wss 3574 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-in 3581 df-ss 3588 |
This theorem is referenced by: cotr3 13717 supcvg 14588 prodfclim1 14625 ef0lem 14809 1strbas 15980 restid 16094 cayley 17834 gsumval3 18308 gsumzaddlem 18321 kgencn3 21361 hmeores 21574 opnfbas 21646 tsmsf1o 21948 ust0 22023 icchmeo 22740 plyeq0lem 23966 ulmdvlem1 24154 basellem7 24813 basellem9 24815 dchrisumlem3 25180 structvtxvallem 25909 struct2griedg 25920 ivthALT 32330 aomclem4 37627 hashnzfzclim 38521 binomcxplemrat 38549 climsuselem1 39839 |
Copyright terms: Public domain | W3C validator |