Theorem djuin 10036

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 4109	. 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ((inl “ 𝐴) ∩ (inr “ 𝐵))
2		imassrn 6216	. . . 4 ⊢ (inr “ 𝐵) ⊆ ran inr
3		djurf1o 10031	. . . . 5 ⊢ inr:V⟶1-1onto({1o} × V)
4		f1of 6854	. . . . 5 ⊢ (inr:V⟶1-1onto({1o} × V) → inr:V⟶({1o} × V))
5		frn 6747	. . . . 5 ⊢ (inr:V⟶({1o} × V) → ran inr ⊆ ({1o} × V))
6	3, 4, 5	mp2b 10	. . . 4 ⊢ ran inr ⊆ ({1o} × V)
7	2, 6	sstri 3922	. . 3 ⊢ (inr “ 𝐵) ⊆ ({1o} × V)
8		incom 4109	. . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = (({1o} × V) ∩ (inl “ 𝐴))
9		imassrn 6216	. . . . . 6 ⊢ (inl “ 𝐴) ⊆ ran inl
10		djulf1o 10030	. . . . . . 7 ⊢ inl:V⟶1-1onto({∅} × V)
11		f1of 6854	. . . . . . 7 ⊢ (inl:V⟶1-1onto({∅} × V) → inl:V⟶({∅} × V))
12		frn 6747	. . . . . . 7 ⊢ (inl:V⟶({∅} × V) → ran inl ⊆ ({∅} × V))
13	10, 11, 12	mp2b 10	. . . . . 6 ⊢ ran inl ⊆ ({∅} × V)
14	9, 13	sstri 3922	. . . . 5 ⊢ (inl “ 𝐴) ⊆ ({∅} × V)
15		1n0 8533	. . . . . . 7 ⊢ 1o ≠ ∅
16	15	necomi 3013	. . . . . 6 ⊢ ∅ ≠ 1o
17		disjsn2 4674	. . . . . 6 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
18		xpdisj1 6170	. . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × V) ∩ ({1o} × V)) = ∅)
19	16, 17, 18	mp2b 10	. . . . 5 ⊢ (({∅} × V) ∩ ({1o} × V)) = ∅
20		ssdisj 4416	. . . . 5 ⊢ (((inl “ 𝐴) ⊆ ({∅} × V) ∧ (({∅} × V) ∩ ({1o} × V)) = ∅) → ((inl “ 𝐴) ∩ ({1o} × V)) = ∅)
21	14, 19, 20	mp2an 705	. . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = ∅
22	8, 21	eqtr3i 2788	. . 3 ⊢ (({1o} × V) ∩ (inl “ 𝐴)) = ∅
23		ssdisj 4416	. . 3 ⊢ (((inr “ 𝐵) ⊆ ({1o} × V) ∧ (({1o} × V) ∩ (inl “ 𝐴)) = ∅) → ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅)
24	7, 22, 23	mp2an 705	. 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅
25	1, 24	eqtr3i 2788	1 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅

This proof depends on syntax axioms:   = wceq 1570   ≠ wne 2959  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  {csn 4588   × cxp 5656  ran crn 5659   “ cima 5661  ⟶wf 6562  ⟶1-1ontowf1o 6565  1_oc1o 8507  inlcinl 10017  inrcinr 10018

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-om 7912  df-1st 8035  df-2nd 8036  df-1o 8514  df-inl 10020  df-inr 10021

This theorem is used by: (None)