Theorem djuin 9871

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 4172	. 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ((inl “ 𝐴) ∩ (inr “ 𝐵))
2		imassrn 6042	. . . 4 ⊢ (inr “ 𝐵) ⊆ ran inr
3		djurf1o 9866	. . . . 5 ⊢ inr:V–1-1-onto→({1o} × V)
4		f1of 6800	. . . . 5 ⊢ (inr:V–1-1-onto→({1o} × V) → inr:V⟶({1o} × V))
5		frn 6695	. . . . 5 ⊢ (inr:V⟶({1o} × V) → ran inr ⊆ ({1o} × V))
6	3, 4, 5	mp2b 10	. . . 4 ⊢ ran inr ⊆ ({1o} × V)
7	2, 6	sstri 3956	. . 3 ⊢ (inr “ 𝐵) ⊆ ({1o} × V)
8		incom 4172	. . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = (({1o} × V) ∩ (inl “ 𝐴))
9		imassrn 6042	. . . . . 6 ⊢ (inl “ 𝐴) ⊆ ran inl
10		djulf1o 9865	. . . . . . 7 ⊢ inl:V–1-1-onto→({∅} × V)
11		f1of 6800	. . . . . . 7 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V))
12		frn 6695	. . . . . . 7 ⊢ (inl:V⟶({∅} × V) → ran inl ⊆ ({∅} × V))
13	10, 11, 12	mp2b 10	. . . . . 6 ⊢ ran inl ⊆ ({∅} × V)
14	9, 13	sstri 3956	. . . . 5 ⊢ (inl “ 𝐴) ⊆ ({∅} × V)
15		1n0 8452	. . . . . . 7 ⊢ 1o ≠ ∅
16	15	necomi 2979	. . . . . 6 ⊢ ∅ ≠ 1o
17		disjsn2 4676	. . . . . 6 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
18		xpdisj1 6134	. . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × V) ∩ ({1o} × V)) = ∅)
19	16, 17, 18	mp2b 10	. . . . 5 ⊢ (({∅} × V) ∩ ({1o} × V)) = ∅
20		ssdisj 4423	. . . . 5 ⊢ (((inl “ 𝐴) ⊆ ({∅} × V) ∧ (({∅} × V) ∩ ({1o} × V)) = ∅) → ((inl “ 𝐴) ∩ ({1o} × V)) = ∅)
21	14, 19, 20	mp2an 692	. . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = ∅
22	8, 21	eqtr3i 2754	. . 3 ⊢ (({1o} × V) ∩ (inl “ 𝐴)) = ∅
23		ssdisj 4423	. . 3 ⊢ (((inr “ 𝐵) ⊆ ({1o} × V) ∧ (({1o} × V) ∩ (inl “ 𝐴)) = ∅) → ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅)
24	7, 22, 23	mp2an 692	. 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅
25	1, 24	eqtr3i 2754	1 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅

Syntax hints:   = wceq 1540   ≠ wne 2925  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  {csn 4589   × cxp 5636  ran crn 5639   “ cima 5641  ⟶wf 6507  –1-1-onto→wf1o 6510  1_oc1o 8427  inlcinl 9852  inrcinr 9853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-1st 7968  df-2nd 7969  df-1o 8434  df-inl 9855  df-inr 9856

This theorem is referenced by: (None)