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Theorem pw1un 4163
 Description: Unit power class distributes over union. (Contributed by SF, 22-Jan-2015.)
Assertion
Ref Expression
pw1un 1(AB) = (1A1B)

Proof of Theorem pw1un
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexun 3443 . . 3 (y (AB)x = {y} ↔ (y A x = {y} y B x = {y}))
2 elpw1 4144 . . 3 (x 1(AB) ↔ y (AB)x = {y})
3 elun 3220 . . . 4 (x (1A1B) ↔ (x 1A x 1B))
4 elpw1 4144 . . . . 5 (x 1Ay A x = {y})
5 elpw1 4144 . . . . 5 (x 1By B x = {y})
64, 5orbi12i 507 . . . 4 ((x 1A x 1B) ↔ (y A x = {y} y B x = {y}))
73, 6bitri 240 . . 3 (x (1A1B) ↔ (y A x = {y} y B x = {y}))
81, 2, 73bitr4i 268 . 2 (x 1(AB) ↔ x (1A1B))
98eqriv 2350 1 1(AB) = (1A1B)
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∪ cun 3207  {csn 3737  ℘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-1c 4136  df-pw1 4137 This theorem is referenced by:  pw1equn  4331  pw1eqadj  4332  ncfinraise  4481  tfindi  4496  tfinsuc  4498  sfindbl  4530  tcdi  6164  ce0addcnnul  6179  ce2  6192
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