Theorem djuin 6810

Description: The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.)

Assertion

Ref	Expression
djuin	|- ( (inl " A ) i^i (inr " B ) ) = (/)

Proof of Theorem djuin

Step	Hyp	Ref	Expression
1		incom 3193	. 2 |- ( (inr " B ) i^i (inl " A ) ) = ( (inl " A ) i^i (inr " B ) )
2		imassrn 4798	. . . 4 |- (inr " B ) C_ ran inr
3		djurf1o 6805	. . . . 5 |- inr : _V -1-1-onto-> ( { 1o } X. _V )
4		f1of 5266	. . . . 5 |- (inr : _V -1-1-onto-> ( { 1o } X. _V ) -> inr : _V --> ( { 1o } X. _V ) )
5		frn 5182	. . . . 5 |- (inr : _V --> ( { 1o } X. _V ) -> ran inr C_ ( { 1o } X. _V ) )
6	3, 4, 5	mp2b 8	. . . 4 |- ran inr C_ ( { 1o } X. _V )
7	2, 6	sstri 3035	. . 3 |- (inr " B ) C_ ( { 1o } X. _V )
8		incom 3193	. . . 4 |- ( (inl " A ) i^i ( { 1o } X. _V ) ) = ( ( { 1o } X. _V ) i^i (inl " A ) )
9		imassrn 4798	. . . . . 6 |- (inl " A ) C_ ran inl
10		djulf1o 6804	. . . . . . 7 |- inl : _V -1-1-onto-> ( { (/) } X. _V )
11		f1of 5266	. . . . . . 7 |- (inl : _V -1-1-onto-> ( { (/) } X. _V ) -> inl : _V --> ( { (/) } X. _V ) )
12		frn 5182	. . . . . . 7 |- (inl : _V --> ( { (/) } X. _V ) -> ran inl C_ ( { (/) } X. _V ) )
13	10, 11, 12	mp2b 8	. . . . . 6 |- ran inl C_ ( { (/) } X. _V )
14	9, 13	sstri 3035	. . . . 5 |- (inl " A ) C_ ( { (/) } X. _V )
15		1n0 6211	. . . . . . 7 |- 1o =/= (/)
16	15	necomi 2341	. . . . . 6 |- (/) =/= 1o
17		disjsn2 3509	. . . . . 6 |- ( (/) =/= 1o -> ( { (/) } i^i { 1o } ) = (/) )
18		xpdisj1 4868	. . . . . 6 |- ( ( { (/) } i^i { 1o } ) = (/) -> ( ( { (/) } X. _V ) i^i ( { 1o } X. _V ) ) = (/) )
19	16, 17, 18	mp2b 8	. . . . 5 |- ( ( { (/) } X. _V ) i^i ( { 1o } X. _V ) ) = (/)
20		ssdisj 3343	. . . . 5 |- ( ( (inl " A ) C_ ( { (/) } X. _V ) /\ ( ( { (/) } X. _V ) i^i ( { 1o } X. _V ) ) = (/) ) -> ( (inl " A ) i^i ( { 1o } X. _V ) ) = (/) )
21	14, 19, 20	mp2an 418	. . . 4 |- ( (inl " A ) i^i ( { 1o } X. _V ) ) = (/)
22	8, 21	eqtr3i 2111	. . 3 |- ( ( { 1o } X. _V ) i^i (inl " A ) ) = (/)
23		ssdisj 3343	. . 3 |- ( ( (inr " B ) C_ ( { 1o } X. _V ) /\ ( ( { 1o } X. _V ) i^i (inl " A ) ) = (/) ) -> ( (inr " B ) i^i (inl " A ) ) = (/) )
24	7, 22, 23	mp2an 418	. 2 |- ( (inr " B ) i^i (inl " A ) ) = (/)
25	1, 24	eqtr3i 2111	1 |- ( (inl " A ) i^i (inr " B ) ) = (/)

Syntax hints:   = wceq 1290   =/= wne 2256   _Vcvv 2620   i^i cin 2999   C_ wss 3000   (/)c0 3287   {csn 3450   X. cxp 4450   ran crn 4453   "cima 4455   -->wf 5024   -1-1-onto->wf1o 5027   1oc1o 6188  inlcinl 6791  inrcinr 6792

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-1st 5925  df-2nd 5926  df-1o 6195  df-inl 6793  df-inr 6794

This theorem is referenced by:  caseinl  6836  dju1p1e2  6884