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Mirrors > Home > MPE Home > Th. List > eqoreldif | Structured version Visualization version Unicode version |
Description: An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.) (Proof shortened by JJ, 23-Jul-2021.) |
Ref | Expression |
---|---|
eqoreldif |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 473 | . . . . 5 | |
2 | elsni 4194 | . . . . . . 7 | |
3 | 2 | con3i 150 | . . . . . 6 |
4 | 3 | adantl 482 | . . . . 5 |
5 | 1, 4 | eldifd 3585 | . . . 4 |
6 | 5 | ex 450 | . . 3 |
7 | 6 | orrd 393 | . 2 |
8 | eleq1a 2696 | . . 3 | |
9 | eldifi 3732 | . . . 4 | |
10 | 9 | a1i 11 | . . 3 |
11 | 8, 10 | jaod 395 | . 2 |
12 | 7, 11 | impbid2 216 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 wceq 1483 wcel 1990 cdif 3571 csn 4177 |
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-v 3202 df-dif 3577 df-sn 4178 |
This theorem is referenced by: lcmfunsnlem2 15353 |
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