Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ssdifim | Structured version Visualization version Unicode version |
Description: Implication of a class difference with a subclass. (Contributed by AV, 3-Jan-2022.) |
Ref | Expression |
---|---|
ssdifim |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss4 3858 | . . 3 | |
2 | eqcom 2629 | . . 3 | |
3 | 1, 2 | sylbb 209 | . 2 |
4 | difeq2 3722 | . . 3 | |
5 | 4 | eqcomd 2628 | . 2 |
6 | 3, 5 | sylan9eq 2676 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 cdif 3571 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-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 |
This theorem is referenced by: ssdifsym 3863 frgrwopregbsn 27181 |
Copyright terms: Public domain | W3C validator |