Theorem pw1ndom3lem 16609

Description: Lemma for pw1ndom3 16610. (Contributed by Jim Kingdon, 14-Feb-2026.)

Hypotheses

Ref	Expression
pw1ndom3lem.x	|- ( ph -> X e. ~P 1o )
pw1ndom3lem.y	|- ( ph -> Y e. ~P 1o )
pw1ndom3lem.z	|- ( ph -> Z e. ~P 1o )
pw1ndom3lem.xy	|- ( ph -> X =/= Y )
pw1ndom3lem.xz	|- ( ph -> X =/= Z )
pw1ndom3lem.yz	|- ( ph -> Y =/= Z )

Assertion

Ref	Expression
pw1ndom3lem	|- ( ph -> X = (/) )

Proof of Theorem pw1ndom3lem

Step	Hyp	Ref	Expression
1		pw1ndom3lem.x	. . . 4 |- ( ph -> X e. ~P 1o )
2	1	elpwid 3663	. . 3 |- ( ph -> X C_ 1o )
3		df1o2 6596	. . 3 |- 1o = { (/) }
4	2, 3	sseqtrdi 3275	. 2 |- ( ph -> X C_ { (/) } )
5		pw1ndom3lem.y	. . . . . . . . . 10 |- ( ph -> Y e. ~P 1o )
6	5	elpwid 3663	. . . . . . . . 9 |- ( ph -> Y C_ 1o )
7	6	adantr 276	. . . . . . . 8 |- ( ( ph /\ X = 1o ) -> Y C_ 1o )
8	7, 3	sseqtrdi 3275	. . . . . . 7 |- ( ( ph /\ X = 1o ) -> Y C_ { (/) } )
9		pw1ndom3lem.xy	. . . . . . . . . . 11 |- ( ph -> X =/= Y )
10	9	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ X = 1o ) -> X =/= Y )
11		neeq1 2415	. . . . . . . . . . 11 |- ( X = 1o -> ( X =/= Y <-> 1o =/= Y ) )
12	11	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ X = 1o ) -> ( X =/= Y <-> 1o =/= Y ) )
13	10, 12	mpbid 147	. . . . . . . . 9 |- ( ( ph /\ X = 1o ) -> 1o =/= Y )
14	13	necomd 2488	. . . . . . . 8 |- ( ( ph /\ X = 1o ) -> Y =/= 1o )
15	3	a1i 9	. . . . . . . 8 |- ( ( ph /\ X = 1o ) -> 1o = { (/) } )
16	14, 15	neeqtrd 2430	. . . . . . 7 |- ( ( ph /\ X = 1o ) -> Y =/= { (/) } )
17		pwntru 4289	. . . . . . 7 |- ( ( Y C_ { (/) } /\ Y =/= { (/) } ) -> Y = (/) )
18	8, 16, 17	syl2anc 411	. . . . . 6 |- ( ( ph /\ X = 1o ) -> Y = (/) )
19		pw1ndom3lem.z	. . . . . . . . . 10 |- ( ph -> Z e. ~P 1o )
20	19	elpwid 3663	. . . . . . . . 9 |- ( ph -> Z C_ 1o )
21	20	adantr 276	. . . . . . . 8 |- ( ( ph /\ X = 1o ) -> Z C_ 1o )
22	21, 3	sseqtrdi 3275	. . . . . . 7 |- ( ( ph /\ X = 1o ) -> Z C_ { (/) } )
23		pw1ndom3lem.xz	. . . . . . . . . . 11 |- ( ph -> X =/= Z )
24	23	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ X = 1o ) -> X =/= Z )
25		neeq1 2415	. . . . . . . . . . 11 |- ( X = 1o -> ( X =/= Z <-> 1o =/= Z ) )
26	25	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ X = 1o ) -> ( X =/= Z <-> 1o =/= Z ) )
27	24, 26	mpbid 147	. . . . . . . . 9 |- ( ( ph /\ X = 1o ) -> 1o =/= Z )
28	27	necomd 2488	. . . . . . . 8 |- ( ( ph /\ X = 1o ) -> Z =/= 1o )
29	28, 15	neeqtrd 2430	. . . . . . 7 |- ( ( ph /\ X = 1o ) -> Z =/= { (/) } )
30		pwntru 4289	. . . . . . 7 |- ( ( Z C_ { (/) } /\ Z =/= { (/) } ) -> Z = (/) )
31	22, 29, 30	syl2anc 411	. . . . . 6 |- ( ( ph /\ X = 1o ) -> Z = (/) )
32	18, 31	eqtr4d 2267	. . . . 5 |- ( ( ph /\ X = 1o ) -> Y = Z )
33		pw1ndom3lem.yz	. . . . . . 7 |- ( ph -> Y =/= Z )
34	33	adantr 276	. . . . . 6 |- ( ( ph /\ X = 1o ) -> Y =/= Z )
35	34	neneqd 2423	. . . . 5 |- ( ( ph /\ X = 1o ) -> -. Y = Z )
36	32, 35	pm2.65da 667	. . . 4 |- ( ph -> -. X = 1o )
37	36	neqned 2409	. . 3 |- ( ph -> X =/= 1o )
38	3	a1i 9	. . 3 |- ( ph -> 1o = { (/) } )
39	37, 38	neeqtrd 2430	. 2 |- ( ph -> X =/= { (/) } )
40		pwntru 4289	. 2 |- ( ( X C_ { (/) } /\ X =/= { (/) } ) -> X = (/) )
41	4, 39, 40	syl2anc 411	1 |- ( ph -> X = (/) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   =/= wne 2402   C_ wss 3200   (/)c0 3494   ~Pcpw 3652   {csn 3669   1oc1o 6575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-suc 4468  df-1o 6582

This theorem is referenced by:  pw1ndom3  16610