Theorem djune 7241

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 6576	. . . . 5 ⊢ 1o ≠ ∅
2	1	nesymi 2446	. . . 4 ⊢ ¬ ∅ = 1o
3		1stinl 7237	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (1st ‘(inl‘𝐴)) = ∅)
4		1stinr 7239	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (1st ‘(inr‘𝐵)) = 1o)
5	3, 4	eqeqan12d 2245	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)) ↔ ∅ = 1o))
6	2, 5	mtbiri 679	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
7		fveq2 5626	. . 3 ⊢ ((inl‘𝐴) = (inr‘𝐵) → (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
8	6, 7	nsyl 631	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (inl‘𝐴) = (inr‘𝐵))
9	8	neqned 2407	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  ∅c0 3491  ‘cfv 5317  1^st c1st 6282  1_oc1o 6553  inlcinl 7208  inrcinr 7209

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fv 5325  df-1st 6284  df-1o 6560  df-inl 7210  df-inr 7211

This theorem is referenced by:  omp1eomlem  7257  difinfsnlem  7262  difinfsn  7263  fodjuomnilemdc  7307  exmidfodomrlemr  7376  exmidfodomrlemrALT  7377