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Theorem ex-pw 29178
Description: Example for df-pw 4561. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
Assertion
Ref Expression
ex-pw (𝐴 = {3, 5, 7} → 𝒫 𝐴 = (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}})))

Proof of Theorem ex-pw
StepHypRef Expression
1 pweq 4573 . 2 (𝐴 = {3, 5, 7} → 𝒫 𝐴 = 𝒫 {3, 5, 7})
2 qdass 4713 . . . 4 ({∅, {3}} ∪ {{5}, {3, 5}}) = ({∅, {3}, {5}} ∪ {{3, 5}})
3 qdassr 4714 . . . 4 ({{7}, {3, 7}} ∪ {{5, 7}, {3, 5, 7}}) = ({{7}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}})
42, 3uneq12i 4120 . . 3 (({∅, {3}} ∪ {{5}, {3, 5}}) ∪ ({{7}, {3, 7}} ∪ {{5, 7}, {3, 5, 7}})) = (({∅, {3}, {5}} ∪ {{3, 5}}) ∪ ({{7}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}}))
5 pwtp 4859 . . 3 𝒫 {3, 5, 7} = (({∅, {3}} ∪ {{5}, {3, 5}}) ∪ ({{7}, {3, 7}} ∪ {{5, 7}, {3, 5, 7}}))
6 df-tp 4590 . . . . . . . 8 {{3}, {5}, {7}} = ({{3}, {5}} ∪ {{7}})
76uneq2i 4119 . . . . . . 7 ({∅} ∪ {{3}, {5}, {7}}) = ({∅} ∪ ({{3}, {5}} ∪ {{7}}))
8 unass 4125 . . . . . . 7 (({∅} ∪ {{3}, {5}}) ∪ {{7}}) = ({∅} ∪ ({{3}, {5}} ∪ {{7}}))
97, 8eqtr4i 2769 . . . . . 6 ({∅} ∪ {{3}, {5}, {7}}) = (({∅} ∪ {{3}, {5}}) ∪ {{7}})
10 tpass 4712 . . . . . . 7 {∅, {3}, {5}} = ({∅} ∪ {{3}, {5}})
1110uneq1i 4118 . . . . . 6 ({∅, {3}, {5}} ∪ {{7}}) = (({∅} ∪ {{3}, {5}}) ∪ {{7}})
129, 11eqtr4i 2769 . . . . 5 ({∅} ∪ {{3}, {5}, {7}}) = ({∅, {3}, {5}} ∪ {{7}})
13 unass 4125 . . . . . 6 (({{3, 5}} ∪ {{3, 7}, {5, 7}}) ∪ {{3, 5, 7}}) = ({{3, 5}} ∪ ({{3, 7}, {5, 7}} ∪ {{3, 5, 7}}))
14 tpass 4712 . . . . . . 7 {{3, 5}, {3, 7}, {5, 7}} = ({{3, 5}} ∪ {{3, 7}, {5, 7}})
1514uneq1i 4118 . . . . . 6 ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}) = (({{3, 5}} ∪ {{3, 7}, {5, 7}}) ∪ {{3, 5, 7}})
16 df-tp 4590 . . . . . . 7 {{3, 7}, {5, 7}, {3, 5, 7}} = ({{3, 7}, {5, 7}} ∪ {{3, 5, 7}})
1716uneq2i 4119 . . . . . 6 ({{3, 5}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}}) = ({{3, 5}} ∪ ({{3, 7}, {5, 7}} ∪ {{3, 5, 7}}))
1813, 15, 173eqtr4i 2776 . . . . 5 ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}) = ({{3, 5}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}})
1912, 18uneq12i 4120 . . . 4 (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}})) = (({∅, {3}, {5}} ∪ {{7}}) ∪ ({{3, 5}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}}))
20 un4 4128 . . . 4 (({∅, {3}, {5}} ∪ {{3, 5}}) ∪ ({{7}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}})) = (({∅, {3}, {5}} ∪ {{7}}) ∪ ({{3, 5}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}}))
2119, 20eqtr4i 2769 . . 3 (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}})) = (({∅, {3}, {5}} ∪ {{3, 5}}) ∪ ({{7}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}}))
224, 5, 213eqtr4i 2776 . 2 𝒫 {3, 5, 7} = (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}))
231, 22eqtrdi 2794 1 (𝐴 = {3, 5, 7} → 𝒫 𝐴 = (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}})))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  cun 3907  c0 4281  𝒫 cpw 4559  {csn 4585  {cpr 4587  {ctp 4589  3c3 12143  5c5 12145  7c7 12147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590
This theorem is referenced by: (None)
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