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Mirrors > Home > MPE Home > Th. List > ssunsn2 | Structured version Visualization version Unicode version |
Description: The property of being sandwiched between two sets naturally splits under union with a singleton. This is the induction hypothesis for the determination of large powersets such as pwtp 4431. (Contributed by Mario Carneiro, 2-Jul-2016.) |
Ref | Expression |
---|---|
ssunsn2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4339 | . . . . 5 | |
2 | unss 3787 | . . . . . . 7 | |
3 | 2 | bicomi 214 | . . . . . 6 |
4 | 3 | rbaibr 946 | . . . . 5 |
5 | 1, 4 | syl 17 | . . . 4 |
6 | 5 | anbi1d 741 | . . 3 |
7 | 2 | biimpi 206 | . . . . . . 7 |
8 | 7 | expcom 451 | . . . . . 6 |
9 | 1, 8 | syl 17 | . . . . 5 |
10 | ssun3 3778 | . . . . . 6 | |
11 | 10 | a1i 11 | . . . . 5 |
12 | 9, 11 | anim12d 586 | . . . 4 |
13 | pm4.72 920 | . . . 4 | |
14 | 12, 13 | sylib 208 | . . 3 |
15 | 6, 14 | bitrd 268 | . 2 |
16 | disjsn 4246 | . . . . . . 7 | |
17 | disj3 4021 | . . . . . . 7 | |
18 | 16, 17 | bitr3i 266 | . . . . . 6 |
19 | sseq1 3626 | . . . . . 6 | |
20 | 18, 19 | sylbi 207 | . . . . 5 |
21 | uncom 3757 | . . . . . . 7 | |
22 | 21 | sseq2i 3630 | . . . . . 6 |
23 | ssundif 4052 | . . . . . 6 | |
24 | 22, 23 | bitr3i 266 | . . . . 5 |
25 | 20, 24 | syl6rbbr 279 | . . . 4 |
26 | 25 | anbi2d 740 | . . 3 |
27 | 3 | simplbi 476 | . . . . . . 7 |
28 | 27 | a1i 11 | . . . . . 6 |
29 | 25 | biimpd 219 | . . . . . 6 |
30 | 28, 29 | anim12d 586 | . . . . 5 |
31 | pm4.72 920 | . . . . 5 | |
32 | 30, 31 | sylib 208 | . . . 4 |
33 | orcom 402 | . . . 4 | |
34 | 32, 33 | syl6bb 276 | . . 3 |
35 | 26, 34 | bitrd 268 | . 2 |
36 | 15, 35 | pm2.61i 176 | 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 cun 3572 cin 3573 wss 3574 c0 3915 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-3an 1039 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-ss 3588 df-nul 3916 df-sn 4178 |
This theorem is referenced by: ssunsn 4360 ssunpr 4365 sstp 4367 |
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