Theorem hashtpglem 11214

Description: Lemma for hashtpg 11215. 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 9306	. . . . . 6 |- 2 e. RR
4		2lt3 9407	. . . . . 6 |- 2 < 3
5	3, 4	ltneii 8369	. . . . 5 |- 2 =/= 3
6	5	neii 2414	. . . 4 |- -. 2 = 3
7		simpr 110	. . . . . . . . 9 |- ( ( ph /\ B = C ) -> B = C )
8	7	tpeq3d 3781	. . . . . . . 8 |- ( ( ph /\ B = C ) -> { A , B , B } = { A , B , C } )
9		tpidm23 3791	. . . . . . . 8 |- { A , B , B } = { A , B }
10	8, 9	eqtr3di 2280	. . . . . . 7 |- ( ( ph /\ B = C ) -> { A , B , C } = { A , B } )
11	10	fveq2d 5673	. . . . . 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 8272	. . . . . . . . . . . 12 |- 1 e. RR
14		1lt3 9408	. . . . . . . . . . . 12 |- 1 < 3
15	13, 14	ltneii 8369	. . . . . . . . . . 11 |- 1 =/= 3
16	15	neii 2414	. . . . . . . . . 10 |- -. 1 = 3
17	10	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , B , C } = { A , B } )
18		dfsn2 3702	. . . . . . . . . . . . . . 15 |- { A } = { A , A }
19		simpr 110	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ B = C ) /\ A = B ) -> A = B )
20	19	preq2d 3774	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , A } = { A , B } )
21	18, 20	eqtrid 2277	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A } = { A , B } )
22	17, 21	eqtr4d 2268	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , B , C } = { A } )
23	22	fveq2d 5673	. . . . . . . . . . . 12 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A , B , C } ) = (♯ ` { A } ) )
24		hashtpglem.a	. . . . . . . . . . . . . 14 |- ( ph -> A e. U )
25		hashsng 11159	. . . . . . . . . . . . . 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 2265	. . . . . . . . . . 11 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A , B , C } ) = 1 )
29	28	eqeq1d 2241	. . . . . . . . . 10 |- ( ( ( ph /\ B = C ) /\ A = B ) -> ( (♯ ` { A , B , C } ) = 3 <-> 1 = 3 ) )
30	16, 29	mtbiri 682	. . . . . . . . 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 2419	. . . . . . 7 |- ( ( ph /\ B = C ) -> A =/= B )
33		hashtpglem.b	. . . . . . . . 9 |- ( ph -> B e. V )
34		hashprg 11171	. . . . . . . . 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 2265	. . . . 5 |- ( ( ph /\ B = C ) -> (♯ ` { A , B , C } ) = 2 )
39	38	eqeq1d 2241	. . . 4 |- ( ( ph /\ B = C ) -> ( (♯ ` { A , B , C } ) = 3 <-> 2 = 3 ) )
40	6, 39	mtbiri 682	. . 3 |- ( ( ph /\ B = C ) -> -. (♯ ` { A , B , C } ) = 3 )
41	2, 40	pm2.65da 667	. 2 |- ( ph -> -. B = C )
42	41	neqned 2419	1 |- ( ph -> B =/= C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   =/= wne 2412   {csn 3688   {cpr 3689   {ctp 3690   ` cfv 5351   1c1 8127   2c2 9287   3c3 9288  ♯chash 11136

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-addcom 8226  ax-addass 8228  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-0id 8234  ax-rnegex 8235  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-tp 3696  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-irdg 6600  df-frec 6621  df-1o 6646  df-oadd 6650  df-er 6766  df-en 6975  df-dom 6976  df-fin 6977  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-inn 9237  df-2 9295  df-3 9296  df-n0 9496  df-z 9577  df-uz 9853  df-fz 10342  df-ihash 11137

This theorem is referenced by:  hashtpg  11215