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Theorem pw1equn 4331

Description: A condition for a unit power class to equal a union. (Contributed by SF, 26-Jan-2015.)

Hypotheses

Ref	Expression
pw1equn.1
pw1equn.2

Assertion

Ref	Expression
pw1equn	1 1 1

Distinct variable groups:   ,,   ,,   ,,

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Proof of Theorem pw1equn

Step	Hyp	Ref	Expression
1		unipw1 4325	. . . 4 1
2		unieq 3900	. . . 4 1 1
3	1, 2	syl5eqr 2399	. . 3 1
4		ssun1 3426	. . . . . 6
5		sseq2 3293	. . . . . 6 1 1
6	4, 5	mpbiri 224	. . . . 5 1 1
7		pw1ss1c 4158	. . . . 5 1 1c
8	6, 7	syl6ss 3284	. . . 4 1 1c
9		eqpw1uni 4330	. . . 4 1c 1
10	8, 9	syl 15	. . 3 1 1
11		ssun2 3427	. . . . . 6
12		sseq2 3293	. . . . . 6 1 1
13	11, 12	mpbiri 224	. . . . 5 1 1
14	13, 7	syl6ss 3284	. . . 4 1 1c
15		eqpw1uni 4330	. . . 4 1c 1
16	14, 15	syl 15	. . 3 1 1
17		pw1equn.1	. . . . 5
18	17	uniex 4317	. . . 4
19		pw1equn.2	. . . . 5
20	19	uniex 4317	. . . 4
21		uneq12 3413	. . . . . . 7
22		uniun 3910	. . . . . . 7
23	21, 22	syl6eqr 2403	. . . . . 6
24	23	eqeq2d 2364	. . . . 5
25		pw1eq 4143	. . . . . . 7 1 1
26	25	eqeq2d 2364	. . . . . 6 1 1
27	26	adantr 451	. . . . 5 1 1
28		pw1eq 4143	. . . . . . 7 1 1
29	28	eqeq2d 2364	. . . . . 6 1 1
30	29	adantl 452	. . . . 5 1 1
31	24, 27, 30	3anbi123d 1252	. . . 4 1 1 1 1
32	18, 20, 31	spc2ev 2947	. . 3 1 1 1 1
33	3, 10, 16, 32	syl3anc 1182	. 2 1 1 1
34		pw1un 4163	. . . 4 1 1 1
35		pw1eq 4143	. . . . . 6 1 1
36		uneq12 3413	. . . . . 6 1 1 1 1
37	35, 36	eqeqan12d 2368	. . . . 5 1 1 1 1 1 1
38	37	3impb 1147	. . . 4 1 1 1 1 1 1
39	34, 38	mpbiri 224	. . 3 1 1 1
40	39	exlimivv 1635	. 2 1 1 1
41	33, 40	impbii 180	1 1 1 1

Colors of variables: wff setvar class

Syntax hints:   wb 176   wa 358   w3a 934  wex 1541   wceq 1642   wcel 1710  cvv 2859   cun 3207   wss 3257  cuni 3891  1cc1c 4134  1 cpw1 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-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  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-ral 2619  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-pr 3742  df-uni 3892  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-imak 4189  df-p6 4191  df-sik 4192  df-ssetk 4193

This theorem is referenced by:  taddc  6229  letc  6231

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