Theorem djune 7137

Description: Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.)

Assertion

Ref	Expression
djune	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))

Proof of Theorem djune

Step	Hyp	Ref	Expression
1		1n0 6485	. . . . 5 ⊢ 1o ≠ ∅
2	1	nesymi 2410	. . . 4 ⊢ ¬ ∅ = 1o
3		1stinl 7133	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (1st ‘(inl‘𝐴)) = ∅)
4		1stinr 7135	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (1st ‘(inr‘𝐵)) = 1o)
5	3, 4	eqeqan12d 2209	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)) ↔ ∅ = 1o))
6	2, 5	mtbiri 676	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
7		fveq2 5554	. . 3 ⊢ ((inl‘𝐴) = (inr‘𝐵) → (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
8	6, 7	nsyl 629	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (inl‘𝐴) = (inr‘𝐵))
9	8	neqned 2371	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164   ≠ wne 2364  ∅c0 3446  ‘cfv 5254  1^st c1st 6191  1_oc1o 6462  inlcinl 7104  inrcinr 7105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fv 5262  df-1st 6193  df-1o 6469  df-inl 7106  df-inr 7107

This theorem is referenced by:  omp1eomlem  7153  difinfsnlem  7158  difinfsn  7159  fodjuomnilemdc  7203  exmidfodomrlemr  7262  exmidfodomrlemrALT  7263