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Theorem ex-pw 27748
Description: Example for df-pw 4319. 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 4320 . 2 (𝐴 = {3, 5, 7} → 𝒫 𝐴 = 𝒫 {3, 5, 7})
2 qdass 4445 . . . 4 ({∅, {3}} ∪ {{5}, {3, 5}}) = ({∅, {3}, {5}} ∪ {{3, 5}})
3 qdassr 4446 . . . 4 ({{7}, {3, 7}} ∪ {{5, 7}, {3, 5, 7}}) = ({{7}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}})
42, 3uneq12i 3929 . . 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 4591 . . 3 𝒫 {3, 5, 7} = (({∅, {3}} ∪ {{5}, {3, 5}}) ∪ ({{7}, {3, 7}} ∪ {{5, 7}, {3, 5, 7}}))
6 df-tp 4341 . . . . . . . 8 {{3}, {5}, {7}} = ({{3}, {5}} ∪ {{7}})
76uneq2i 3928 . . . . . . 7 ({∅} ∪ {{3}, {5}, {7}}) = ({∅} ∪ ({{3}, {5}} ∪ {{7}}))
8 unass 3934 . . . . . . 7 (({∅} ∪ {{3}, {5}}) ∪ {{7}}) = ({∅} ∪ ({{3}, {5}} ∪ {{7}}))
97, 8eqtr4i 2790 . . . . . 6 ({∅} ∪ {{3}, {5}, {7}}) = (({∅} ∪ {{3}, {5}}) ∪ {{7}})
10 tpass 4444 . . . . . . 7 {∅, {3}, {5}} = ({∅} ∪ {{3}, {5}})
1110uneq1i 3927 . . . . . 6 ({∅, {3}, {5}} ∪ {{7}}) = (({∅} ∪ {{3}, {5}}) ∪ {{7}})
129, 11eqtr4i 2790 . . . . 5 ({∅} ∪ {{3}, {5}, {7}}) = ({∅, {3}, {5}} ∪ {{7}})
13 unass 3934 . . . . . 6 (({{3, 5}} ∪ {{3, 7}, {5, 7}}) ∪ {{3, 5, 7}}) = ({{3, 5}} ∪ ({{3, 7}, {5, 7}} ∪ {{3, 5, 7}}))
14 tpass 4444 . . . . . . 7 {{3, 5}, {3, 7}, {5, 7}} = ({{3, 5}} ∪ {{3, 7}, {5, 7}})
1514uneq1i 3927 . . . . . 6 ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}) = (({{3, 5}} ∪ {{3, 7}, {5, 7}}) ∪ {{3, 5, 7}})
16 df-tp 4341 . . . . . . 7 {{3, 7}, {5, 7}, {3, 5, 7}} = ({{3, 7}, {5, 7}} ∪ {{3, 5, 7}})
1716uneq2i 3928 . . . . . 6 ({{3, 5}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}}) = ({{3, 5}} ∪ ({{3, 7}, {5, 7}} ∪ {{3, 5, 7}}))
1813, 15, 173eqtr4i 2797 . . . . 5 ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}) = ({{3, 5}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}})
1912, 18uneq12i 3929 . . . 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 3937 . . . 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 2790 . . 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 2797 . 2 𝒫 {3, 5, 7} = (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}))
231, 22syl6eq 2815 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 1652  cun 3732  c0 4081  𝒫 cpw 4317  {csn 4336  {cpr 4338  {ctp 4340  3c3 11330  5c5 11332  7c7 11334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341
This theorem is referenced by: (None)
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