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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 (A 1cA = 1A)

Proof of Theorem eqpw1uni
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3267 . . 3 (A 1c → (x Ax 1c))
2 pw1ss1c 4158 . . . . 5 1A 1c
32sseli 3269 . . . 4 (x 1Ax 1c)
43a1i 10 . . 3 (A 1c → (x 1Ax 1c))
5 el1c 4139 . . . 4 (x 1cy x = {y})
6 vex 2862 . . . . . . . . . 10 y V
76snid 3760 . . . . . . . . 9 y {y}
8 eleq2 2414 . . . . . . . . . 10 (x = {y} → (y xy {y}))
98rspcev 2955 . . . . . . . . 9 (({y} A y {y}) → x A y x)
107, 9mpan2 652 . . . . . . . 8 ({y} Ax A y x)
11 el1c 4139 . . . . . . . . . . 11 (x 1cz x = {z})
12 elsn 3748 . . . . . . . . . . . . . . 15 (y {z} ↔ y = z)
13 sneq 3744 . . . . . . . . . . . . . . . 16 (y = z → {y} = {z})
1413eleq1d 2419 . . . . . . . . . . . . . . 15 (y = z → ({y} A ↔ {z} A))
1512, 14sylbi 187 . . . . . . . . . . . . . 14 (y {z} → ({y} A ↔ {z} A))
1615biimprcd 216 . . . . . . . . . . . . 13 ({z} A → (y {z} → {y} A))
17 eleq1 2413 . . . . . . . . . . . . . 14 (x = {z} → (x A ↔ {z} A))
18 eleq2 2414 . . . . . . . . . . . . . . 15 (x = {z} → (y xy {z}))
1918imbi1d 308 . . . . . . . . . . . . . 14 (x = {z} → ((y x → {y} A) ↔ (y {z} → {y} A)))
2017, 19imbi12d 311 . . . . . . . . . . . . 13 (x = {z} → ((x A → (y x → {y} A)) ↔ ({z} A → (y {z} → {y} A))))
2116, 20mpbiri 224 . . . . . . . . . . . 12 (x = {z} → (x A → (y x → {y} A)))
2221exlimiv 1634 . . . . . . . . . . 11 (z x = {z} → (x A → (y x → {y} A)))
2311, 22sylbi 187 . . . . . . . . . 10 (x 1c → (x A → (y x → {y} A)))
241, 23syli 33 . . . . . . . . 9 (A 1c → (x A → (y x → {y} A)))
2524rexlimdv 2737 . . . . . . . 8 (A 1c → (x A y x → {y} A))
2610, 25impbid2 195 . . . . . . 7 (A 1c → ({y} Ax A y x))
27 eluni2 3895 . . . . . . 7 (y Ax A y x)
2826, 27syl6bbr 254 . . . . . 6 (A 1c → ({y} Ay A))
29 eleq1 2413 . . . . . . 7 (x = {y} → (x A ↔ {y} A))
30 eleq1 2413 . . . . . . . 8 (x = {y} → (x 1A ↔ {y} 1A))
31 snelpw1 4146 . . . . . . . 8 ({y} 1Ay A)
3230, 31syl6bb 252 . . . . . . 7 (x = {y} → (x 1Ay A))
3329, 32bibi12d 312 . . . . . 6 (x = {y} → ((x Ax 1A) ↔ ({y} Ay A)))
3428, 33syl5ibrcom 213 . . . . 5 (A 1c → (x = {y} → (x Ax 1A)))
3534exlimdv 1636 . . . 4 (A 1c → (y x = {y} → (x Ax 1A)))
365, 35syl5bi 208 . . 3 (A 1c → (x 1c → (x Ax 1A)))
371, 4, 36pm5.21ndd 343 . 2 (A 1c → (x Ax 1A))
3837eqrdv 2351 1 (A 1cA = 1A)
 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  ℘1cpw1 4135 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  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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