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Theorem ssunsn2 3958
 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 4012. (Contributed by Mario Carneiro, 2-Jul-2016.)
Assertion
Ref Expression
ssunsn2

Proof of Theorem ssunsn2
StepHypRef Expression
1 snssi 3942 . . . . 5
2 unss 3521 . . . . . . 7
32bicomi 194 . . . . . 6
43rbaibr 875 . . . . 5
51, 4syl 16 . . . 4
65anbi1d 686 . . 3
72biimpi 187 . . . . . . 7
87expcom 425 . . . . . 6
91, 8syl 16 . . . . 5
10 ssun3 3512 . . . . . 6
1110a1i 11 . . . . 5
129, 11anim12d 547 . . . 4
13 pm4.72 847 . . . 4
1412, 13sylib 189 . . 3
156, 14bitrd 245 . 2
16 disjsn 3868 . . . . . . 7
17 disj3 3672 . . . . . . 7
1816, 17bitr3i 243 . . . . . 6
19 sseq1 3369 . . . . . 6
2018, 19sylbi 188 . . . . 5
21 uncom 3491 . . . . . . 7
2221sseq2i 3373 . . . . . 6
23 ssundif 3711 . . . . . 6
2422, 23bitr3i 243 . . . . 5
2520, 24syl6rbbr 256 . . . 4
2625anbi2d 685 . . 3
273simplbi 447 . . . . . . 7
2827a1i 11 . . . . . 6
2925biimpd 199 . . . . . 6
3028, 29anim12d 547 . . . . 5
31 pm4.72 847 . . . . 5
3230, 31sylib 189 . . . 4
33 orcom 377 . . . 4
3432, 33syl6bb 253 . . 3
3526, 34bitrd 245 . 2
3615, 35pm2.61i 158 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359   wceq 1652   wcel 1725   cdif 3317   cun 3318   cin 3319   wss 3320  c0 3628  csn 3814 This theorem is referenced by:  ssunsn  3959  ssunpr  3961  sstp  3963 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-sn 3820
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