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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 6461 | . . . . 5 ⊢ 1o ≠ ∅ | |
2 | 1 | nesymi 2406 | . . . 4 ⊢ ¬ ∅ = 1o |
3 | 1stinl 7107 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (1st ‘(inl‘𝐴)) = ∅) | |
4 | 1stinr 7109 | . . . . 5 ⊢ (𝐵 ∈ 𝑊 → (1st ‘(inr‘𝐵)) = 1o) | |
5 | 3, 4 | eqeqan12d 2205 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵)) ↔ ∅ = 1o)) |
6 | 2, 5 | mtbiri 676 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ¬ (1st ‘(inl‘𝐴)) = (1st ‘(inr‘𝐵))) |
7 | fveq2 5537 | . . 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 2367 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (inl‘𝐴) ≠ (inr‘𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 ≠ wne 2360 ∅c0 3437 ‘cfv 5238 1st c1st 6167 1oc1o 6438 inlcinl 7078 inrcinr 7079 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-br 4022 df-opab 4083 df-mpt 4084 df-tr 4120 df-id 4314 df-iord 4387 df-on 4389 df-suc 4392 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-iota 5199 df-fun 5240 df-fv 5246 df-1st 6169 df-1o 6445 df-inl 7080 df-inr 7081 |
This theorem is referenced by: omp1eomlem 7127 difinfsnlem 7132 difinfsn 7133 fodjuomnilemdc 7177 exmidfodomrlemr 7236 exmidfodomrlemrALT 7237 |
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