Theorem djune 7195

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 6531	. . . . 5 ⊢ 1o ≠ ∅
2	1	nesymi 2423	. . . 4 ⊢ ¬ ∅ = 1o
3		1stinl 7191	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (1st ‘(inl‘𝐴)) = ∅)
4		1stinr 7193	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (1st ‘(inr‘𝐵)) = 1o)
5	3, 4	eqeqan12d 2222	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)) ↔ ∅ = 1o))
6	2, 5	mtbiri 677	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)))
7		fveq2 5589	. . 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 2384	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177   ≠ wne 2377  ∅c0 3464  ‘cfv 5280  1^st c1st 6237  1_oc1o 6508  inlcinl 7162  inrcinr 7163

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-iota 5241  df-fun 5282  df-fv 5288  df-1st 6239  df-1o 6515  df-inl 7164  df-inr 7165

This theorem is referenced by:  omp1eomlem  7211  difinfsnlem  7216  difinfsn  7217  fodjuomnilemdc  7261  exmidfodomrlemr  7326  exmidfodomrlemrALT  7327