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Mirrors > Home > ILE Home > Th. List > djune | GIF version |
Description: Left and right injection never produce equal values. (Contributed by Jim Kingdon, 2-Jul-2022.) |
Ref | Expression |
---|---|
djune | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) ≠ (inr‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1n0 6329 | . . . . 5 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 2354 | . . . 4 ⊢ ¬ ∅ = 1o |
3 | 1stinl 6959 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (1st ‘(inl‘𝐴)) = ∅) | |
4 | 1stinr 6961 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → (1st ‘(inr‘𝐵)) = 1o) | |
5 | 3, 4 | eqeqan12d 2155 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)) ↔ ∅ = 1o)) |
6 | 2, 5 | mtbiri 664 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵))) |
7 | fveq2 5421 | . . 3 ⊢ ((inl‘𝐴) = (inr‘𝐵) → (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵))) | |
8 | 6, 7 | nsyl 617 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (inl‘𝐴) = (inr‘𝐵)) |
9 | 8 | neqned 2315 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) ≠ (inr‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 ∅c0 3363 ‘cfv 5123 1st c1st 6036 1oc1o 6306 inlcinl 6930 inrcinr 6931 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fv 5131 df-1st 6038 df-1o 6313 df-inl 6932 df-inr 6933 |
This theorem is referenced by: omp1eomlem 6979 difinfsnlem 6984 difinfsn 6985 fodjuomnilemdc 7016 exmidfodomrlemr 7058 exmidfodomrlemrALT 7059 |
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