Theorem hashtpglem 11243

Description: Lemma for hashtpg 11244. 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 9324	. . . . . 6 |- 2 e. RR
4		2lt3 9425	. . . . . 6 |- 2 < 3
5	3, 4	ltneii 8386	. . . . 5 |- 2 =/= 3
6	5	neii 2416	. . . 4 |- -. 2 = 3
7		simpr 110	. . . . . . . . 9 |- ( ( ph /\ B = C ) -> B = C )
8	7	tpeq3d 3787	. . . . . . . 8 |- ( ( ph /\ B = C ) -> { A , B , B } = { A , B , C } )
9		tpidm23 3797	. . . . . . . 8 |- { A , B , B } = { A , B }
10	8, 9	eqtr3di 2282	. . . . . . 7 |- ( ( ph /\ B = C ) -> { A , B , C } = { A , B } )
11	10	fveq2d 5679	. . . . . 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 8289	. . . . . . . . . . . 12 |- 1 e. RR
14		1lt3 9426	. . . . . . . . . . . 12 |- 1 < 3
15	13, 14	ltneii 8386	. . . . . . . . . . 11 |- 1 =/= 3
16	15	neii 2416	. . . . . . . . . 10 |- -. 1 = 3
17	10	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , B , C } = { A , B } )
18		dfsn2 3708	. . . . . . . . . . . . . . 15 |- { A } = { A , A }
19		simpr 110	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ B = C ) /\ A = B ) -> A = B )
20	19	preq2d 3780	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , A } = { A , B } )
21	18, 20	eqtrid 2279	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A } = { A , B } )
22	17, 21	eqtr4d 2270	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , B , C } = { A } )
23	22	fveq2d 5679	. . . . . . . . . . . 12 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A , B , C } ) = (♯ ` { A } ) )
24		hashtpglem.a	. . . . . . . . . . . . . 14 |- ( ph -> A e. U )
25		hashsng 11186	. . . . . . . . . . . . . 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 2267	. . . . . . . . . . 11 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A , B , C } ) = 1 )
29	28	eqeq1d 2243	. . . . . . . . . 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 2421	. . . . . . 7 |- ( ( ph /\ B = C ) -> A =/= B )
33		hashtpglem.b	. . . . . . . . 9 |- ( ph -> B e. V )
34		hashprg 11198	. . . . . . . . 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 2267	. . . . 5 |- ( ( ph /\ B = C ) -> (♯ ` { A , B , C } ) = 2 )
39	38	eqeq1d 2243	. . . 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 2421	1 |- ( ph -> B =/= C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   =/= wne 2414   {csn 3694   {cpr 3695   {ctp 3696   ` cfv 5357   1c1 8144   2c2 9305   3c3 9306  ♯chash 11163

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-tp 3702  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-ihash 11164

This theorem is referenced by:  hashtpg  11244