Theorem djuin 9903

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) ∩ (inr “ 𝐵)) = ∅

Proof of Theorem djuin

Step	Hyp	Ref	Expression
1		incom 4170	. 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ((inl “ 𝐴) ∩ (inr “ 𝐵))
2		imassrn 6074	. . . 4 ⊢ (inr “ 𝐵) ⊆ ran inr
3		djurf1o 9898	. . . . 5 ⊢ inr:V–1-1-onto→({1o} × V)
4		f1of 6821	. . . . 5 ⊢ (inr:V–1-1-onto→({1o} × V) → inr:V⟶({1o} × V))
5		frn 6714	. . . . 5 ⊢ (inr:V⟶({1o} × V) → ran inr ⊆ ({1o} × V))
6	3, 4, 5	mp2b 10	. . . 4 ⊢ ran inr ⊆ ({1o} × V)
7	2, 6	sstri 3954	. . 3 ⊢ (inr “ 𝐵) ⊆ ({1o} × V)
8		incom 4170	. . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = (({1o} × V) ∩ (inl “ 𝐴))
9		imassrn 6074	. . . . . 6 ⊢ (inl “ 𝐴) ⊆ ran inl
10		djulf1o 9897	. . . . . . 7 ⊢ inl:V–1-1-onto→({∅} × V)
11		f1of 6821	. . . . . . 7 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V))
12		frn 6714	. . . . . . 7 ⊢ (inl:V⟶({∅} × V) → ran inl ⊆ ({∅} × V))
13	10, 11, 12	mp2b 10	. . . . . 6 ⊢ ran inl ⊆ ({∅} × V)
14	9, 13	sstri 3954	. . . . 5 ⊢ (inl “ 𝐴) ⊆ ({∅} × V)
15		1n0 8471	. . . . . . 7 ⊢ 1o ≠ ∅
16	15	necomi 3018	. . . . . 6 ⊢ ∅ ≠ 1o
17		disjsn2 4683	. . . . . 6 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
18		xpdisj1 6159	. . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × V) ∩ ({1o} × V)) = ∅)
19	16, 17, 18	mp2b 10	. . . . 5 ⊢ (({∅} × V) ∩ ({1o} × V)) = ∅
20		ssdisj 4426	. . . . 5 ⊢ (((inl “ 𝐴) ⊆ ({∅} × V) ∧ (({∅} × V) ∩ ({1o} × V)) = ∅) → ((inl “ 𝐴) ∩ ({1o} × V)) = ∅)
21	14, 19, 20	mp2an 704	. . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = ∅
22	8, 21	eqtr3i 2794	. . 3 ⊢ (({1o} × V) ∩ (inl “ 𝐴)) = ∅
23		ssdisj 4426	. . 3 ⊢ (((inr “ 𝐵) ⊆ ({1o} × V) ∧ (({1o} × V) ∩ (inl “ 𝐴)) = ∅) → ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅)
24	7, 22, 23	mp2an 704	. 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅
25	1, 24	eqtr3i 2794	1 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅

Syntax hints:   = wceq 1567   ≠ wne 2964  Vcvv 3463   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  {csn 4594   × cxp 5660  ran crn 5663   “ cima 5665  ⟶wf 6533  –1-1-onto→wf1o 6536  1_oc1o 8445  inlcinl 9884  inrcinr 9885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7862  df-1st 7985  df-2nd 7986  df-1o 8452  df-inl 9887  df-inr 9888

This theorem is referenced by: (None)