Theorem hashtpglem 11300

Description: Lemma for hashtpg 11301. 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 9375	. . . . . 6 |- 2 e. RR
4		2lt3 9477	. . . . . 6 |- 2 < 3
5	3, 4	ltneii 8422	. . . . 5 |- 2 =/= 3
6	5	neii 2422	. . . 4 |- -. 2 = 3
7		simpr 110	. . . . . . . . 9 |- ( ( ph /\ B = C ) -> B = C )
8	7	tpeq3d 3802	. . . . . . . 8 |- ( ( ph /\ B = C ) -> { A , B , B } = { A , B , C } )
9		tpidm23 3812	. . . . . . . 8 |- { A , B , B } = { A , B }
10	8, 9	eqtr3di 2286	. . . . . . 7 |- ( ( ph /\ B = C ) -> { A , B , C } = { A , B } )
11	10	fveq2d 5699	. . . . . 6 |- ( ( ph /\ B = C ) -> (♯ ` { A , B , C } ) = (♯ ` { A , B } ) )
12	1	ad2antrr 492	. . . . . . . . 9 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A , B , C } ) = 3 )
13		1re 8325	. . . . . . . . . . . 12 |- 1 e. RR
14		1lt3 9478	. . . . . . . . . . . 12 |- 1 < 3
15	13, 14	ltneii 8422	. . . . . . . . . . 11 |- 1 =/= 3
16	15	neii 2422	. . . . . . . . . 10 |- -. 1 = 3
17	10	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , B , C } = { A , B } )
18		dfsn2 3723	. . . . . . . . . . . . . . 15 |- { A } = { A , A }
19		simpr 110	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ B = C ) /\ A = B ) -> A = B )
20	19	preq2d 3795	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , A } = { A , B } )
21	18, 20	eqtrid 2283	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A } = { A , B } )
22	17, 21	eqtr4d 2274	. . . . . . . . . . . . 13 |- ( ( ( ph /\ B = C ) /\ A = B ) -> { A , B , C } = { A } )
23	22	fveq2d 5699	. . . . . . . . . . . 12 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A , B , C } ) = (♯ ` { A } ) )
24		hashtpglem.a	. . . . . . . . . . . . . 14 |- ( ph -> A e. U )
25		hashsng 11239	. . . . . . . . . . . . . 14 |- ( A e. U -> (♯ ` { A } ) = 1 )
26	24, 25	syl 14	. . . . . . . . . . . . 13 |- ( ph -> (♯ ` { A } ) = 1 )
27	26	ad2antrr 492	. . . . . . . . . . . 12 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A } ) = 1 )
28	23, 27	eqtrd 2271	. . . . . . . . . . 11 |- ( ( ( ph /\ B = C ) /\ A = B ) -> (♯ ` { A , B , C } ) = 1 )
29	28	eqeq1d 2247	. . . . . . . . . 10 |- ( ( ( ph /\ B = C ) /\ A = B ) -> ( (♯ ` { A , B , C } ) = 3 <-> 1 = 3 ) )
30	16, 29	mtbiri 686	. . . . . . . . 9 |- ( ( ( ph /\ B = C ) /\ A = B ) -> -. (♯ ` { A , B , C } ) = 3 )
31	12, 30	pm2.65da 671	. . . . . . . 8 |- ( ( ph /\ B = C ) -> -. A = B )
32	31	neqned 2427	. . . . . . 7 |- ( ( ph /\ B = C ) -> A =/= B )
33		hashtpglem.b	. . . . . . . . 9 |- ( ph -> B e. V )
34		hashprg 11251	. . . . . . . . 9 |- ( ( A e. U /\ B e. V ) -> ( A =/= B <-> (♯ ` { A , B } ) = 2 ) )
35	24, 33, 34	syl2anc 415	. . . . . . . 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 2271	. . . . 5 |- ( ( ph /\ B = C ) -> (♯ ` { A , B , C } ) = 2 )
39	38	eqeq1d 2247	. . . 4 |- ( ( ph /\ B = C ) -> ( (♯ ` { A , B , C } ) = 3 <-> 2 = 3 ) )
40	6, 39	mtbiri 686	. . 3 |- ( ( ph /\ B = C ) -> -. (♯ ` { A , B , C } ) = 3 )
41	2, 40	pm2.65da 671	. 2 |- ( ph -> -. B = C )
42	41	neqned 2427	1 |- ( ph -> B =/= C )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   {csn 3709   {cpr 3710   {ctp 3711   ` cfv 5377   1c1 8180   2c2 9356   3c3 9357  ♯chash 11216

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9306  df-2 9364  df-3 9365  df-n0 9566  df-z 9647  df-uz 9924  df-fz 10414  df-ihash 11217

This theorem is used by:  hashtpg  11301