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Mirrors > Home > MPE Home > Th. List > djuin | Structured version Visualization version GIF version |
Description: The images of any classes under right and left injection produce disjoint sets. (Contributed by Jim Kingdon, 21-Jun-2022.) |
Ref | Expression |
---|---|
djuin | ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | incom 4128 | . 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ((inl “ 𝐴) ∩ (inr “ 𝐵)) | |
2 | imassrn 5907 | . . . 4 ⊢ (inr “ 𝐵) ⊆ ran inr | |
3 | djurf1o 9326 | . . . . 5 ⊢ inr:V–1-1-onto→({1o} × V) | |
4 | f1of 6590 | . . . . 5 ⊢ (inr:V–1-1-onto→({1o} × V) → inr:V⟶({1o} × V)) | |
5 | frn 6493 | . . . . 5 ⊢ (inr:V⟶({1o} × V) → ran inr ⊆ ({1o} × V)) | |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ ran inr ⊆ ({1o} × V) |
7 | 2, 6 | sstri 3924 | . . 3 ⊢ (inr “ 𝐵) ⊆ ({1o} × V) |
8 | incom 4128 | . . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = (({1o} × V) ∩ (inl “ 𝐴)) | |
9 | imassrn 5907 | . . . . . 6 ⊢ (inl “ 𝐴) ⊆ ran inl | |
10 | djulf1o 9325 | . . . . . . 7 ⊢ inl:V–1-1-onto→({∅} × V) | |
11 | f1of 6590 | . . . . . . 7 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V)) | |
12 | frn 6493 | . . . . . . 7 ⊢ (inl:V⟶({∅} × V) → ran inl ⊆ ({∅} × V)) | |
13 | 10, 11, 12 | mp2b 10 | . . . . . 6 ⊢ ran inl ⊆ ({∅} × V) |
14 | 9, 13 | sstri 3924 | . . . . 5 ⊢ (inl “ 𝐴) ⊆ ({∅} × V) |
15 | 1n0 8102 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
16 | 15 | necomi 3041 | . . . . . 6 ⊢ ∅ ≠ 1o |
17 | disjsn2 4608 | . . . . . 6 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅) | |
18 | xpdisj1 5985 | . . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × V) ∩ ({1o} × V)) = ∅) | |
19 | 16, 17, 18 | mp2b 10 | . . . . 5 ⊢ (({∅} × V) ∩ ({1o} × V)) = ∅ |
20 | ssdisj 4367 | . . . . 5 ⊢ (((inl “ 𝐴) ⊆ ({∅} × V) ∧ (({∅} × V) ∩ ({1o} × V)) = ∅) → ((inl “ 𝐴) ∩ ({1o} × V)) = ∅) | |
21 | 14, 19, 20 | mp2an 691 | . . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = ∅ |
22 | 8, 21 | eqtr3i 2823 | . . 3 ⊢ (({1o} × V) ∩ (inl “ 𝐴)) = ∅ |
23 | ssdisj 4367 | . . 3 ⊢ (((inr “ 𝐵) ⊆ ({1o} × V) ∧ (({1o} × V) ∩ (inl “ 𝐴)) = ∅) → ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅) | |
24 | 7, 22, 23 | mp2an 691 | . 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅ |
25 | 1, 24 | eqtr3i 2823 | 1 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ≠ wne 2987 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 × cxp 5517 ran crn 5520 “ cima 5522 ⟶wf 6320 –1-1-onto→wf1o 6323 1oc1o 8078 inlcinl 9312 inrcinr 9313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-om 7561 df-1st 7671 df-2nd 7672 df-1o 8085 df-inl 9315 df-inr 9316 |
This theorem is referenced by: (None) |
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