Theorem djuin 9340

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 4171	. 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ((inl “ 𝐴) ∩ (inr “ 𝐵))
2		imassrn 5933	. . . 4 ⊢ (inr “ 𝐵) ⊆ ran inr
3		djurf1o 9335	. . . . 5 ⊢ inr:V–1-1-onto→({1o} × V)
4		f1of 6608	. . . . 5 ⊢ (inr:V–1-1-onto→({1o} × V) → inr:V⟶({1o} × V))
5		frn 6513	. . . . 5 ⊢ (inr:V⟶({1o} × V) → ran inr ⊆ ({1o} × V))
6	3, 4, 5	mp2b 10	. . . 4 ⊢ ran inr ⊆ ({1o} × V)
7	2, 6	sstri 3969	. . 3 ⊢ (inr “ 𝐵) ⊆ ({1o} × V)
8		incom 4171	. . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = (({1o} × V) ∩ (inl “ 𝐴))
9		imassrn 5933	. . . . . 6 ⊢ (inl “ 𝐴) ⊆ ran inl
10		djulf1o 9334	. . . . . . 7 ⊢ inl:V–1-1-onto→({∅} × V)
11		f1of 6608	. . . . . . 7 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V))
12		frn 6513	. . . . . . 7 ⊢ (inl:V⟶({∅} × V) → ran inl ⊆ ({∅} × V))
13	10, 11, 12	mp2b 10	. . . . . 6 ⊢ ran inl ⊆ ({∅} × V)
14	9, 13	sstri 3969	. . . . 5 ⊢ (inl “ 𝐴) ⊆ ({∅} × V)
15		1n0 8112	. . . . . . 7 ⊢ 1o ≠ ∅
16	15	necomi 3069	. . . . . 6 ⊢ ∅ ≠ 1o
17		disjsn2 4641	. . . . . 6 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
18		xpdisj1 6011	. . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × V) ∩ ({1o} × V)) = ∅)
19	16, 17, 18	mp2b 10	. . . . 5 ⊢ (({∅} × V) ∩ ({1o} × V)) = ∅
20		ssdisj 4402	. . . . 5 ⊢ (((inl “ 𝐴) ⊆ ({∅} × V) ∧ (({∅} × V) ∩ ({1o} × V)) = ∅) → ((inl “ 𝐴) ∩ ({1o} × V)) = ∅)
21	14, 19, 20	mp2an 690	. . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = ∅
22	8, 21	eqtr3i 2845	. . 3 ⊢ (({1o} × V) ∩ (inl “ 𝐴)) = ∅
23		ssdisj 4402	. . 3 ⊢ (((inr “ 𝐵) ⊆ ({1o} × V) ∧ (({1o} × V) ∩ (inl “ 𝐴)) = ∅) → ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅)
24	7, 22, 23	mp2an 690	. 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅
25	1, 24	eqtr3i 2845	1 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅

Syntax hints:   = wceq 1536   ≠ wne 3015  Vcvv 3491   ∩ cin 3928   ⊆ wss 3929  ∅c0 4284  {csn 4560   × cxp 5546  ran crn 5549   “ cima 5551  ⟶wf 6344  –1-1-onto→wf1o 6347  1_oc1o 8088  inlcinl 9321  inrcinr 9322

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5323  ax-un 7454

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-ne 3016  df-ral 3142  df-rex 3143  df-rab 3146  df-v 3493  df-sbc 3769  df-dif 3932  df-un 3934  df-in 3936  df-ss 3945  df-pss 3947  df-nul 4285  df-if 4461  df-pw 4534  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-om 7574  df-1st 7682  df-2nd 7683  df-1o 8095  df-inl 9324  df-inr 9325

This theorem is referenced by: (None)