Theorem hashtpglem 11111

Description: Lemma for hashtpg 11112. This is one of the three not-equal conclusions required for the reverse direction. (Contributed by Jim Kingdon, 18-Apr-2026.)

Hypotheses

Ref	Expression
hashtpglem.a	|- ( ph -> A e. U )
hashtpglem.b	|- ( ph -> B e. V )
hashtpglem.c	|- ( ph -> C e. W )
hashtpglem.3	|- ( ph -> (♯ ` { A , B , C } ) = 3 )

Assertion

Ref	Expression
hashtpglem	|- ( ph -> B =/= C )

Proof of Theorem hashtpglem

Step	Hyp	Ref	Expression
1		hashtpglem.3	. . . 4 |- ( ph -> (♯ ` { A , B , C } ) = 3 )
2	1	adantr 276	. . 3 |- ( ( ph /\ B = C ) -> (♯ ` { A , B , C } ) = 3 )
3		2re 9213	. . . . . 6 |- 2 e. RR
4		2lt3 9314	. . . . . 6 |- 2 < 3
5	3, 4	ltneii 8276	. . . . 5 |- 2 =/= 3
6	5	neii 2404	. . . 4 |- -. 2 = 3
7		simpr 110	. . . . . . . . 9 |- ( ( ph /\ B = C ) -> B = C )
8	7	tpeq3d 3762	. . . . . . . 8 |- ( ( ph /\ B = C ) -> { A , B , B } = { A , B , C } )
9		tpidm23 3772	. . . . . . . 8 |- { A , B , B } = { A , B }
10	8, 9	eqtr3di 2279	. . . . . . 7 |- ( ( ph /\ B = C ) -> { A , B , C } = { A , B } )
11	10	fveq2d 5643	. . . . . 6 |- ( ( ph /\ B = C ) -> (♯ ` { A , B , C } ) = (♯ ` { A , B } ) )
12	1	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A , B , C } ) = 3 )
13		1re 8178	. . . . . . . . . . . 12 |- 1 e. RR
14		1lt3 9315	. . . . . . . . . . . 12 |- 1 < 3
15	13, 14	ltneii 8276	. . . . . . . . . . 11 |- 1 =/= 3
16	15	neii 2404	. . . . . . . . . 10 |- -. 1 = 3
17	10	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , B , C } = { A , B } )
18		dfsn2 3683	. . . . . . . . . . . . . . 15 |- { A } = { A , A }
19		simpr 110	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ B = C ) /\ A = B ) -> A = B )
20	19	preq2d 3755	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , A } = { A , B } )
21	18, 20	eqtrid 2276	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A } = { A , B } )
22	17, 21	eqtr4d 2267	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , B , C } = { A } )
23	22	fveq2d 5643	. . . . . . . . . . . 12 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A , B , C } ) = (♯ ` { A } ) )
24		hashtpglem.a	. . . . . . . . . . . . . 14 |- ( ph -> A e. U )
25		hashsng 11061	. . . . . . . . . . . . . 14 |- ( A e. U -> (♯ ` { A } ) = 1 )
26	24, 25	syl 14	. . . . . . . . . . . . 13 |- ( ph -> (♯ ` { A } ) = 1 )
27	26	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A } ) = 1 )
28	23, 27	eqtrd 2264	. . . . . . . . . . 11 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A , B , C } ) = 1 )
29	28	eqeq1d 2240	. . . . . . . . . 10 |- ( ( ( ph /\ B = C ) /\ A = B ) -> ( (♯ ` { A , B , C } ) = 3 <-> 1 = 3 ) )
30	16, 29	mtbiri 681	. . . . . . . . 9 |- ( ( ( ph /\ B = C ) /\ A = B ) -> -. (♯ ` { A , B , C } ) = 3 )
31	12, 30	pm2.65da 667	. . . . . . . 8 |- ( ( ph /\ B = C ) -> -. A = B )
32	31	neqned 2409	. . . . . . 7 |- ( ( ph /\ B = C ) -> A =/= B )
33		hashtpglem.b	. . . . . . . . 9 |- ( ph -> B e. V )
34		hashprg 11073	. . . . . . . . 9 |- ( ( A e. U /\ B e. V ) -> ( A =/= B <-> (♯ ` { A , B } ) = 2 ) )
35	24, 33, 34	syl2anc 411	. . . . . . . 8 |- ( ph -> ( A =/= B <-> (♯ ` { A , B } ) = 2 ) )
36	35	adantr 276	. . . . . . 7 |- ( ( ph /\ B = C ) -> ( A =/= B <-> (♯ ` { A , B } ) = 2 ) )
37	32, 36	mpbid 147	. . . . . 6 |- ( ( ph /\ B = C ) -> (♯ ` { A , B } ) = 2 )
38	11, 37	eqtrd 2264	. . . . 5 |- ( ( ph /\ B = C ) -> (♯ ` { A , B , C } ) = 2 )
39	38	eqeq1d 2240	. . . 4 |- ( ( ph /\ B = C ) -> ( (♯ ` { A , B , C } ) = 3 <-> 2 = 3 ) )
40	6, 39	mtbiri 681	. . 3 |- ( ( ph /\ B = C ) -> -. (♯ ` { A , B , C } ) = 3 )
41	2, 40	pm2.65da 667	. 2 |- ( ph -> -. B = C )
42	41	neqned 2409	1 |- ( ph -> B =/= C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   =/= wne 2402   {csn 3669   {cpr 3670   {ctp 3671   ` cfv 5326   1c1 8033   2c2 9194   3c3 9195  ♯chash 11038

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-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-3 9203  df-n0 9403  df-z 9480  df-uz 9756  df-fz 10244  df-ihash 11039

This theorem is referenced by:  hashtpg  11112