Theorem hashtpglem 11154

Description: Lemma for hashtpg 11155. 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 9256	. . . . . 6 |- 2 e. RR
4		2lt3 9357	. . . . . 6 |- 2 < 3
5	3, 4	ltneii 8319	. . . . 5 |- 2 =/= 3
6	5	neii 2405	. . . 4 |- -. 2 = 3
7		simpr 110	. . . . . . . . 9 |- ( ( ph /\ B = C ) -> B = C )
8	7	tpeq3d 3766	. . . . . . . 8 |- ( ( ph /\ B = C ) -> { A , B , B } = { A , B , C } )
9		tpidm23 3776	. . . . . . . 8 |- { A , B , B } = { A , B }
10	8, 9	eqtr3di 2279	. . . . . . 7 |- ( ( ph /\ B = C ) -> { A , B , C } = { A , B } )
11	10	fveq2d 5652	. . . . . 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 8221	. . . . . . . . . . . 12 |- 1 e. RR
14		1lt3 9358	. . . . . . . . . . . 12 |- 1 < 3
15	13, 14	ltneii 8319	. . . . . . . . . . 11 |- 1 =/= 3
16	15	neii 2405	. . . . . . . . . 10 |- -. 1 = 3
17	10	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , B , C } = { A , B } )
18		dfsn2 3687	. . . . . . . . . . . . . . 15 |- { A } = { A , A }
19		simpr 110	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ B = C ) /\ A = B ) -> A = B )
20	19	preq2d 3759	. . . . . . . . . . . . . . 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 5652	. . . . . . . . . . . 12 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A , B , C } ) = (♯ ` { A } ) )
24		hashtpglem.a	. . . . . . . . . . . . . 14 |- ( ph -> A e. U )
25		hashsng 11104	. . . . . . . . . . . . . 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 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 2410	. . . . . . 7 |- ( ( ph /\ B = C ) -> A =/= B )
33		hashtpglem.b	. . . . . . . . 9 |- ( ph -> B e. V )
34		hashprg 11116	. . . . . . . . 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 682	. . 3 |- ( ( ph /\ B = C ) -> -. (♯ ` { A , B , C } ) = 3 )
41	2, 40	pm2.65da 667	. 2 |- ( ph -> -. B = C )
42	41	neqned 2410	1 |- ( ph -> B =/= C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   =/= wne 2403   {csn 3673   {cpr 3674   {ctp 3675   ` cfv 5333   1c1 8076   2c2 9237   3c3 9238  ♯chash 11081

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8259  df-mnf 8260  df-xr 8261  df-ltxr 8262  df-le 8263  df-sub 8395  df-neg 8396  df-inn 9187  df-2 9245  df-3 9246  df-n0 9446  df-z 9523  df-uz 9799  df-fz 10287  df-ihash 11082

This theorem is referenced by:  hashtpg  11155