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Mirrors > Home > NFE Home > Th. List > nnsucelrlem3 | Unicode version |
Description: Lemma for nnsucelr 4428. Rearrange union and difference for a particular group of classes. (Contributed by SF, 15-Jan-2015.) |
Ref | Expression |
---|---|
nnsucelrlem3.1 |
Ref | Expression |
---|---|
nnsucelrlem3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | indir 3503 | . . . . 5 ∼ ∼ ∼ | |
2 | df-dif 3215 | . . . . . . . 8 ∼ | |
3 | 2 | eqcomi 2357 | . . . . . . 7 ∼ |
4 | incompl 4073 | . . . . . . 7 ∼ | |
5 | 3, 4 | uneq12i 3416 | . . . . . 6 ∼ ∼ |
6 | un0 3575 | . . . . . 6 | |
7 | 5, 6 | eqtri 2373 | . . . . 5 ∼ ∼ |
8 | 1, 7 | eqtri 2373 | . . . 4 ∼ |
9 | difsn 3845 | . . . . 5 | |
10 | 9 | 3ad2ant3 978 | . . . 4 |
11 | 8, 10 | syl5req 2398 | . . 3 ∼ |
12 | simp2 956 | . . . 4 | |
13 | df-ne 2518 | . . . . . . . 8 | |
14 | 13 | biimpi 186 | . . . . . . 7 |
15 | 14 | 3ad2ant1 976 | . . . . . 6 |
16 | nnsucelrlem3.1 | . . . . . . . . 9 | |
17 | 16 | elcompl 3225 | . . . . . . . 8 ∼ |
18 | 16 | elsnc 3756 | . . . . . . . 8 |
19 | 17, 18 | xchbinx 301 | . . . . . . 7 ∼ |
20 | 16 | snss 3838 | . . . . . . 7 ∼ ∼ |
21 | 19, 20 | bitr3i 242 | . . . . . 6 ∼ |
22 | 15, 21 | sylib 188 | . . . . 5 ∼ |
23 | ssequn2 3436 | . . . . 5 ∼ ∼ ∼ | |
24 | 22, 23 | sylib 188 | . . . 4 ∼ ∼ |
25 | 12, 24 | ineq12d 3458 | . . 3 ∼ ∼ |
26 | 11, 25 | eqtr4d 2388 | . 2 ∼ |
27 | df-dif 3215 | . . . 4 ∼ | |
28 | 27 | uneq1i 3414 | . . 3 ∼ |
29 | undir 3504 | . . 3 ∼ ∼ | |
30 | 28, 29 | eqtri 2373 | . 2 ∼ |
31 | 26, 30 | syl6eqr 2403 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 w3a 934 wceq 1642 wcel 1710 wne 2516 cvv 2859 ∼ ccompl 3205 cdif 3206 cun 3207 cin 3208 wss 3257 c0 3550 csn 3737 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-ss 3259 df-nul 3551 df-sn 3741 |
This theorem is referenced by: nnsucelr 4428 |
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