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Mirrors > Home > MPE Home > Th. List > undif4 | Structured version Visualization version Unicode version |
Description: Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
undif4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm2.621 424 | . . . . . . 7 | |
2 | olc 399 | . . . . . . 7 | |
3 | 1, 2 | impbid1 215 | . . . . . 6 |
4 | 3 | anbi2d 740 | . . . . 5 |
5 | eldif 3584 | . . . . . . 7 | |
6 | 5 | orbi2i 541 | . . . . . 6 |
7 | ordi 908 | . . . . . 6 | |
8 | 6, 7 | bitri 264 | . . . . 5 |
9 | elun 3753 | . . . . . 6 | |
10 | 9 | anbi1i 731 | . . . . 5 |
11 | 4, 8, 10 | 3bitr4g 303 | . . . 4 |
12 | elun 3753 | . . . 4 | |
13 | eldif 3584 | . . . 4 | |
14 | 11, 12, 13 | 3bitr4g 303 | . . 3 |
15 | 14 | alimi 1739 | . 2 |
16 | disj1 4019 | . 2 | |
17 | dfcleq 2616 | . 2 | |
18 | 15, 16, 17 | 3imtr4i 281 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wo 383 wa 384 wal 1481 wceq 1483 wcel 1990 cdif 3571 cun 3572 cin 3573 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-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-nul 3916 |
This theorem is referenced by: phplem1 8139 infdifsn 8554 difico 29545 caratheodorylem1 40740 |
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