Theorem ex-pw 27900

Description: Example for df-pw 4455. 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 4456	. 2 ⊢ (𝐴 = {3, 5, 7} → 𝒫 𝐴 = 𝒫 {3, 5, 7})
2		qdass 4596	. . . 4 ⊢ ({∅, {3}} ∪ {{5}, {3, 5}}) = ({∅, {3}, {5}} ∪ {{3, 5}})
3		qdassr 4597	. . . 4 ⊢ ({{7}, {3, 7}} ∪ {{5, 7}, {3, 5, 7}}) = ({{7}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}})
4	2, 3	uneq12i 4058	. . 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 4740	. . 3 ⊢ 𝒫 {3, 5, 7} = (({∅, {3}} ∪ {{5}, {3, 5}}) ∪ ({{7}, {3, 7}} ∪ {{5, 7}, {3, 5, 7}}))
6		df-tp 4477	. . . . . . . 8 ⊢ {{3}, {5}, {7}} = ({{3}, {5}} ∪ {{7}})
7	6	uneq2i 4057	. . . . . . 7 ⊢ ({∅} ∪ {{3}, {5}, {7}}) = ({∅} ∪ ({{3}, {5}} ∪ {{7}}))
8		unass 4063	. . . . . . 7 ⊢ (({∅} ∪ {{3}, {5}}) ∪ {{7}}) = ({∅} ∪ ({{3}, {5}} ∪ {{7}}))
9	7, 8	eqtr4i 2822	. . . . . 6 ⊢ ({∅} ∪ {{3}, {5}, {7}}) = (({∅} ∪ {{3}, {5}}) ∪ {{7}})
10		tpass 4595	. . . . . . 7 ⊢ {∅, {3}, {5}} = ({∅} ∪ {{3}, {5}})
11	10	uneq1i 4056	. . . . . 6 ⊢ ({∅, {3}, {5}} ∪ {{7}}) = (({∅} ∪ {{3}, {5}}) ∪ {{7}})
12	9, 11	eqtr4i 2822	. . . . 5 ⊢ ({∅} ∪ {{3}, {5}, {7}}) = ({∅, {3}, {5}} ∪ {{7}})
13		unass 4063	. . . . . 6 ⊢ (({{3, 5}} ∪ {{3, 7}, {5, 7}}) ∪ {{3, 5, 7}}) = ({{3, 5}} ∪ ({{3, 7}, {5, 7}} ∪ {{3, 5, 7}}))
14		tpass 4595	. . . . . . 7 ⊢ {{3, 5}, {3, 7}, {5, 7}} = ({{3, 5}} ∪ {{3, 7}, {5, 7}})
15	14	uneq1i 4056	. . . . . 6 ⊢ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}) = (({{3, 5}} ∪ {{3, 7}, {5, 7}}) ∪ {{3, 5, 7}})
16		df-tp 4477	. . . . . . 7 ⊢ {{3, 7}, {5, 7}, {3, 5, 7}} = ({{3, 7}, {5, 7}} ∪ {{3, 5, 7}})
17	16	uneq2i 4057	. . . . . 6 ⊢ ({{3, 5}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}}) = ({{3, 5}} ∪ ({{3, 7}, {5, 7}} ∪ {{3, 5, 7}}))
18	13, 15, 17	3eqtr4i 2829	. . . . 5 ⊢ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}) = ({{3, 5}} ∪ {{3, 7}, {5, 7}, {3, 5, 7}})
19	12, 18	uneq12i 4058	. . . 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 4066	. . . 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 2822	. . 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 2829	. 2 ⊢ 𝒫 {3, 5, 7} = (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}}))
23	1, 22	syl6eq 2847	1 ⊢ (𝐴 = {3, 5, 7} → 𝒫 𝐴 = (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}})))

Syntax hints:   → wi 4   = wceq 1522   ∪ cun 3857  ∅c0 4211  𝒫 cpw 4453  {csn 4472  {cpr 4474  {ctp 4476  3c3 11541  5c5 11543  7c7 11545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477

This theorem is referenced by: (None)