Theorem ex-pw 30689

Description: Example for df-pw 4560. 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

Step	Hyp	Ref	Expression
1		pweq 4572	. 2 ⊢ (𝐴 = {3, 5, 7} → 𝒫 𝐴 = 𝒫 {3, 5, 7})
2		qdass 4715	. . . 4 ⊢ ({∅, {3}} ∪ {{5}, {3, 5}}) = ({∅, {3}, {5}} ∪ {{3, 5}})
3		qdassr 4716	. . . 4 ⊢ ({{7}, {3, 7}} ∪ {{5, 7}, {3, 5, 7}}) = ({{7}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}})
4	2, 3	uneq12i 4122	. . 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 4863	. . 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}})
7	6	uneq2i 4121	. . . . . . 7 ⊢ ({∅} ∪ {{3}, {5}, {7}}) = ({∅} ∪ ({{3}, {5}} ∪ {{7}}))
8		unass 4127	. . . . . . 7 ⊢ (({∅} ∪ {{3}, {5}}) ∪ {{7}}) = ({∅} ∪ ({{3}, {5}} ∪ {{7}}))
9	7, 8	eqtr4i 2791	. . . . . 6 ⊢ ({∅} ∪ {{3}, {5}, {7}}) = (({∅} ∪ {{3}, {5}}) ∪ {{7}})
10		tpass 4714	. . . . . . 7 ⊢ {∅, {3}, {5}} = ({∅} ∪ {{3}, {5}})
11	10	uneq1i 4120	. . . . . 6 ⊢ ({∅, {3}, {5}} ∪ {{7}}) = (({∅} ∪ {{3}, {5}}) ∪ {{7}})
12	9, 11	eqtr4i 2791	. . . . 5 ⊢ ({∅} ∪ {{3}, {5}, {7}}) = ({∅, {3}, {5}} ∪ {{7}})
13		unass 4127	. . . . . 6 ⊢ (({{3, 5}} ∪ {{3, 7}, {5, 7}}) ∪ {{3, 5, 7}}) = ({{3, 5}} ∪ ({{3, 7}, {5, 7}} ∪ {{3, 5, 7}}))
14		tpass 4714	. . . . . . 7 ⊢ {{3, 5}, {3, 7}, {5, 7}} = ({{3, 5}} ∪ {{3, 7}, {5, 7}})
15	14	uneq1i 4120	. . . . . 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}})
17	16	uneq2i 4121	. . . . . 6 ⊢ ({{3, 5}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}}) = ({{3, 5}} ∪ ({{3, 7}, {5, 7}} ∪ {{3, 5, 7}}))
18	13, 15, 17	3eqtr4i 2798	. . . . 5 ⊢ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}) = ({{3, 5}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}})
19	12, 18	uneq12i 4122	. . . 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 4130	. . . 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}}))
21	19, 20	eqtr4i 2791	. . 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}}))
22	4, 5, 21	3eqtr4i 2798	. 2 ⊢ 𝒫 {3, 5, 7} = (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}))
23	1, 22	eqtrdi 2816	1 ⊢ (𝐴 = {3, 5, 7} → 𝒫 𝐴 = (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}})))

Syntax hints:   → wi 4   = wceq 1563   ∪ cun 3905  ∅c0 4288  𝒫 cpw 4558  {csn 4585  {cpr 4587  {ctp 4589  3c3 12287  5c5 12289  7c7 12291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590

This theorem is referenced by: (None)