Theorem djuin 9876

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 4161	. 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ((inl “ 𝐴) ∩ (inr “ 𝐵))
2		imassrn 6060	. . . 4 ⊢ (inr “ 𝐵) ⊆ ran inr
3		djurf1o 9871	. . . . 5 ⊢ inr:V–1-1-onto→({1o} × V)
4		f1of 6806	. . . . 5 ⊢ (inr:V–1-1-onto→({1o} × V) → inr:V⟶({1o} × V))
5		frn 6699	. . . . 5 ⊢ (inr:V⟶({1o} × V) → ran inr ⊆ ({1o} × V))
6	3, 4, 5	mp2b 10	. . . 4 ⊢ ran inr ⊆ ({1o} × V)
7	2, 6	sstri 3945	. . 3 ⊢ (inr “ 𝐵) ⊆ ({1o} × V)
8		incom 4161	. . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = (({1o} × V) ∩ (inl “ 𝐴))
9		imassrn 6060	. . . . . 6 ⊢ (inl “ 𝐴) ⊆ ran inl
10		djulf1o 9870	. . . . . . 7 ⊢ inl:V–1-1-onto→({∅} × V)
11		f1of 6806	. . . . . . 7 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V))
12		frn 6699	. . . . . . 7 ⊢ (inl:V⟶({∅} × V) → ran inl ⊆ ({∅} × V))
13	10, 11, 12	mp2b 10	. . . . . 6 ⊢ ran inl ⊆ ({∅} × V)
14	9, 13	sstri 3945	. . . . 5 ⊢ (inl “ 𝐴) ⊆ ({∅} × V)
15		1n0 8456	. . . . . . 7 ⊢ 1o ≠ ∅
16	15	necomi 3011	. . . . . 6 ⊢ ∅ ≠ 1o
17		disjsn2 4671	. . . . . 6 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅)
18		xpdisj1 6146	. . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × V) ∩ ({1o} × V)) = ∅)
19	16, 17, 18	mp2b 10	. . . . 5 ⊢ (({∅} × V) ∩ ({1o} × V)) = ∅
20		ssdisj 4414	. . . . 5 ⊢ (((inl “ 𝐴) ⊆ ({∅} × V) ∧ (({∅} × V) ∩ ({1o} × V)) = ∅) → ((inl “ 𝐴) ∩ ({1o} × V)) = ∅)
21	14, 19, 20	mp2an 702	. . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = ∅
22	8, 21	eqtr3i 2787	. . 3 ⊢ (({1o} × V) ∩ (inl “ 𝐴)) = ∅
23		ssdisj 4414	. . 3 ⊢ (((inr “ 𝐵) ⊆ ({1o} × V) ∧ (({1o} × V) ∩ (inl “ 𝐴)) = ∅) → ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅)
24	7, 22, 23	mp2an 702	. 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅
25	1, 24	eqtr3i 2787	1 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅

Syntax hints:   = wceq 1560   ≠ wne 2957  Vcvv 3454   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4582   × cxp 5645  ran crn 5648   “ cima 5650  ⟶wf 6517  –1-1-onto→wf1o 6520  1_oc1o 8430  inlcinl 9857  inrcinr 9858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-om 7847  df-1st 7970  df-2nd 7971  df-1o 8437  df-inl 9860  df-inr 9861

This theorem is referenced by: (None)