djune - Intuitionistic Logic Explorer

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Theorem djune 7079

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 6435	. . . . 5 ⊢ 1_o ≠ ∅
2	1	nesymi 2393	. . . 4 ⊢ ¬ ∅ = 1_o
3		1stinl 7075	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (1^st ‘(inl‘𝐴)) = ∅)
4		1stinr 7077	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → (1^st ‘(inr‘𝐵)) = 1_o)
5	3, 4	eqeqan12d 2193	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((1^st ‘(inl‘𝐴)) = (1^st ‘(inr‘𝐵)) ↔ ∅ = 1_o))
6	2, 5	mtbiri 675	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (1^st ‘(inl‘𝐴)) = (1^st ‘(inr‘𝐵)))
7		fveq2 5517	. . 3 ⊢ ((inl‘𝐴) = (inr‘𝐵) → (1^st ‘(inl‘𝐴)) = (1^st ‘(inr‘𝐵)))
8	6, 7	nsyl 628	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (inl‘𝐴) = (inr‘𝐵))
9	8	neqned 2354	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) ≠ (inr‘𝐵))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∅c0 3424 ‘cfv 5218 1^st c1st 6141 1_oc1o 6412 inlcinl 7046 inrcinr 7047

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-id 4295 df-iord 4368 df-on 4370 df-suc 4373 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-iota 5180 df-fun 5220 df-fv 5226 df-1st 6143 df-1o 6419 df-inl 7048 df-inr 7049

This theorem is referenced by: omp1eomlem 7095 difinfsnlem 7100 difinfsn 7101 fodjuomnilemdc 7144 exmidfodomrlemr 7203 exmidfodomrlemrALT 7204