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Theorem ssunsn2 4189
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 4246. (Contributed by Mario Carneiro, 2-Jul-2016.)
Assertion
Ref Expression
ssunsn2

Proof of Theorem ssunsn2
StepHypRef Expression
1 snssi 4174 . . . . 5
2 unss 3677 . . . . . . 7
32bicomi 202 . . . . . 6
43rbaibr 905 . . . . 5
51, 4syl 16 . . . 4
65anbi1d 704 . . 3
72biimpi 194 . . . . . . 7
87expcom 435 . . . . . 6
91, 8syl 16 . . . . 5
10 ssun3 3668 . . . . . 6
1110a1i 11 . . . . 5
129, 11anim12d 563 . . . 4
13 pm4.72 876 . . . 4
1412, 13sylib 196 . . 3
156, 14bitrd 253 . 2
16 disjsn 4090 . . . . . . 7
17 disj3 3871 . . . . . . 7
1816, 17bitr3i 251 . . . . . 6
19 sseq1 3524 . . . . . 6
2018, 19sylbi 195 . . . . 5
21 uncom 3647 . . . . . . 7
2221sseq2i 3528 . . . . . 6
23 ssundif 3911 . . . . . 6
2422, 23bitr3i 251 . . . . 5
2520, 24syl6rbbr 264 . . . 4
2625anbi2d 703 . . 3
273simplbi 460 . . . . . . 7
2827a1i 11 . . . . . 6
2925biimpd 207 . . . . . 6
3028, 29anim12d 563 . . . . 5
31 pm4.72 876 . . . . 5
3230, 31sylib 196 . . . 4
33 orcom 387 . . . 4
3432, 33syl6bb 261 . . 3
3526, 34bitrd 253 . 2
3615, 35pm2.61i 164 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  \cdif 3472  u.cun 3473  i^icin 3474  C_wss 3475   c0 3784  {csn 4029
This theorem is referenced by:  ssunsn  4190  ssunpr  4192  sstp  4194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-sn 4030
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