Theorem djune 7055

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 6411	. . . . 5 ⊢ 1o ≠ ∅
2	1	nesymi 2386	. . . 4 ⊢ ¬ ∅ = 1o
3		1stinl 7051	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (1st ‘(inl‘𝐴)) = ∅)
4		1stinr 7053	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (1st ‘(inr‘𝐵)) = 1o)
5	3, 4	eqeqan12d 2186	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)) ↔ ∅ = 1o))
6	2, 5	mtbiri 670	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
7		fveq2 5496	. . 3 ⊢ ((inl‘𝐴) = (inr‘𝐵) → (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
8	6, 7	nsyl 623	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (inl‘𝐴) = (inr‘𝐵))
9	8	neqned 2347	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  ∅c0 3414  ‘cfv 5198  1^st c1st 6117  1_oc1o 6388  inlcinl 7022  inrcinr 7023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-1st 6119  df-1o 6395  df-inl 7024  df-inr 7025

This theorem is referenced by:  omp1eomlem  7071  difinfsnlem  7076  difinfsn  7077  fodjuomnilemdc  7120  exmidfodomrlemr  7179  exmidfodomrlemrALT  7180