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Mirrors > Home > MPE Home > Th. List > Mathboxes > difininv | Structured version Visualization version Unicode version |
Description: Condition for the intersections of two sets with a given set to be equal. (Contributed by Thierry Arnoux, 28-Dec-2021.) |
Ref | Expression |
---|---|
difininv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssdif0 3942 | . . . . . 6 | |
2 | indif1 3871 | . . . . . . 7 | |
3 | 2 | eqeq1i 2627 | . . . . . 6 |
4 | 1, 3 | bitr4i 267 | . . . . 5 |
5 | 4 | biimpri 218 | . . . 4 |
6 | 5 | adantr 481 | . . 3 |
7 | inss2 3834 | . . . 4 | |
8 | 7 | a1i 11 | . . 3 |
9 | 6, 8 | ssind 3837 | . 2 |
10 | ssdif0 3942 | . . . . . 6 | |
11 | indif1 3871 | . . . . . . 7 | |
12 | 11 | eqeq1i 2627 | . . . . . 6 |
13 | 10, 12 | bitr4i 267 | . . . . 5 |
14 | 13 | biimpri 218 | . . . 4 |
15 | 14 | adantl 482 | . . 3 |
16 | inss2 3834 | . . . 4 | |
17 | 16 | a1i 11 | . . 3 |
18 | 15, 17 | ssind 3837 | . 2 |
19 | 9, 18 | eqssd 3620 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 cdif 3571 cin 3573 wss 3574 c0 3915 |
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 df-nul 3916 |
This theorem is referenced by: chtvalz 30707 |
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