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Theorem unipw1 4325
 Description: The union of a unit power class is the original set. (Contributed by SF, 20-Jan-2015.)
Assertion
Ref Expression
unipw1 1A = A

Proof of Theorem unipw1
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni 3894 . . 3 (x 1Ay(x y y 1A))
2 elpw1 4144 . . . . . 6 (y 1Az A y = {z})
32anbi1i 676 . . . . 5 ((y 1A x y) ↔ (z A y = {z} x y))
4 ancom 437 . . . . 5 ((x y y 1A) ↔ (y 1A x y))
5 r19.41v 2764 . . . . 5 (z A (y = {z} x y) ↔ (z A y = {z} x y))
63, 4, 53bitr4i 268 . . . 4 ((x y y 1A) ↔ z A (y = {z} x y))
76exbii 1582 . . 3 (y(x y y 1A) ↔ yz A (y = {z} x y))
8 risset 2661 . . . 4 (x Az A z = x)
9 snex 4111 . . . . . . 7 {z} V
10 eleq2 2414 . . . . . . 7 (y = {z} → (x yx {z}))
119, 10ceqsexv 2894 . . . . . 6 (y(y = {z} x y) ↔ x {z})
12 df-sn 3741 . . . . . . 7 {z} = {x x = z}
1312abeq2i 2460 . . . . . 6 (x {z} ↔ x = z)
14 equcom 1680 . . . . . 6 (x = zz = x)
1511, 13, 143bitri 262 . . . . 5 (y(y = {z} x y) ↔ z = x)
1615rexbii 2639 . . . 4 (z A y(y = {z} x y) ↔ z A z = x)
17 rexcom4 2878 . . . 4 (z A y(y = {z} x y) ↔ yz A (y = {z} x y))
188, 16, 173bitr2ri 265 . . 3 (yz A (y = {z} x y) ↔ x A)
191, 7, 183bitri 262 . 2 (x 1Ax A)
2019eqriv 2350 1 1A = A
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615  {csn 3737  ∪cuni 3891  ℘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-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:  pw1exb  4326  pw1equn  4331  pw1eqadj  4332  sspw1  4335
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