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Theorem eqpw1uni 4330
 Description: A class of singletons is equal to the unit power class of its union. (Contributed by SF, 26-Jan-2015.)
Assertion
Ref Expression
eqpw1uni 1c 1

Proof of Theorem eqpw1uni
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3267 . . 3 1c 1c
2 pw1ss1c 4158 . . . . 5 1 1c
32sseli 3269 . . . 4 1 1c
43a1i 10 . . 3 1c 1 1c
5 el1c 4139 . . . 4 1c
6 vex 2862 . . . . . . . . . 10
76snid 3760 . . . . . . . . 9
8 eleq2 2414 . . . . . . . . . 10
98rspcev 2955 . . . . . . . . 9
107, 9mpan2 652 . . . . . . . 8
11 el1c 4139 . . . . . . . . . . 11 1c
12 elsn 3748 . . . . . . . . . . . . . . 15
13 sneq 3744 . . . . . . . . . . . . . . . 16
1413eleq1d 2419 . . . . . . . . . . . . . . 15
1512, 14sylbi 187 . . . . . . . . . . . . . 14
1615biimprcd 216 . . . . . . . . . . . . 13
17 eleq1 2413 . . . . . . . . . . . . . 14
18 eleq2 2414 . . . . . . . . . . . . . . 15
1918imbi1d 308 . . . . . . . . . . . . . 14
2017, 19imbi12d 311 . . . . . . . . . . . . 13
2116, 20mpbiri 224 . . . . . . . . . . . 12
2221exlimiv 1634 . . . . . . . . . . 11
2311, 22sylbi 187 . . . . . . . . . 10 1c
241, 23syli 33 . . . . . . . . 9 1c
2524rexlimdv 2737 . . . . . . . 8 1c
2610, 25impbid2 195 . . . . . . 7 1c
27 eluni2 3895 . . . . . . 7
2826, 27syl6bbr 254 . . . . . 6 1c
29 eleq1 2413 . . . . . . 7
30 eleq1 2413 . . . . . . . 8 1 1
31 snelpw1 4146 . . . . . . . 8 1
3230, 31syl6bb 252 . . . . . . 7 1
3329, 32bibi12d 312 . . . . . 6 1
3428, 33syl5ibrcom 213 . . . . 5 1c 1
3534exlimdv 1636 . . . 4 1c 1
365, 35syl5bi 208 . . 3 1c 1c 1
371, 4, 36pm5.21ndd 343 . 2 1c 1
3837eqrdv 2351 1 1c 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176  wex 1541   wceq 1642   wcel 1710  wrex 2615   wss 3257  csn 3737  cuni 3891  1cc1c 4134  1 cpw1 4135 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2619  df-rex 2620  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-pw 3724  df-sn 3741  df-uni 3892  df-1c 4136  df-pw1 4137 This theorem is referenced by:  pw1equn  4331  pw1eqadj  4332  sspw1  4335  sspw12  4336
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