![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4193 | . 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ((inl “ 𝐴) ∩ (inr “ 𝐵)) | |
2 | imassrn 6060 | . . . 4 ⊢ (inr “ 𝐵) ⊆ ran inr | |
3 | djurf1o 9904 | . . . . 5 ⊢ inr:V–1-1-onto→({1o} × V) | |
4 | f1of 6823 | . . . . 5 ⊢ (inr:V–1-1-onto→({1o} × V) → inr:V⟶({1o} × V)) | |
5 | frn 6714 | . . . . 5 ⊢ (inr:V⟶({1o} × V) → ran inr ⊆ ({1o} × V)) | |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ ran inr ⊆ ({1o} × V) |
7 | 2, 6 | sstri 3983 | . . 3 ⊢ (inr “ 𝐵) ⊆ ({1o} × V) |
8 | incom 4193 | . . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = (({1o} × V) ∩ (inl “ 𝐴)) | |
9 | imassrn 6060 | . . . . . 6 ⊢ (inl “ 𝐴) ⊆ ran inl | |
10 | djulf1o 9903 | . . . . . . 7 ⊢ inl:V–1-1-onto→({∅} × V) | |
11 | f1of 6823 | . . . . . . 7 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V)) | |
12 | frn 6714 | . . . . . . 7 ⊢ (inl:V⟶({∅} × V) → ran inl ⊆ ({∅} × V)) | |
13 | 10, 11, 12 | mp2b 10 | . . . . . 6 ⊢ ran inl ⊆ ({∅} × V) |
14 | 9, 13 | sstri 3983 | . . . . 5 ⊢ (inl “ 𝐴) ⊆ ({∅} × V) |
15 | 1n0 8483 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
16 | 15 | necomi 2987 | . . . . . 6 ⊢ ∅ ≠ 1o |
17 | disjsn2 4708 | . . . . . 6 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅) | |
18 | xpdisj1 6150 | . . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × V) ∩ ({1o} × V)) = ∅) | |
19 | 16, 17, 18 | mp2b 10 | . . . . 5 ⊢ (({∅} × V) ∩ ({1o} × V)) = ∅ |
20 | ssdisj 4451 | . . . . 5 ⊢ (((inl “ 𝐴) ⊆ ({∅} × V) ∧ (({∅} × V) ∩ ({1o} × V)) = ∅) → ((inl “ 𝐴) ∩ ({1o} × V)) = ∅) | |
21 | 14, 19, 20 | mp2an 689 | . . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = ∅ |
22 | 8, 21 | eqtr3i 2754 | . . 3 ⊢ (({1o} × V) ∩ (inl “ 𝐴)) = ∅ |
23 | ssdisj 4451 | . . 3 ⊢ (((inr “ 𝐵) ⊆ ({1o} × V) ∧ (({1o} × V) ∩ (inl “ 𝐴)) = ∅) → ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅) | |
24 | 7, 22, 23 | mp2an 689 | . 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅ |
25 | 1, 24 | eqtr3i 2754 | 1 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ≠ wne 2932 Vcvv 3466 ∩ cin 3939 ⊆ wss 3940 ∅c0 4314 {csn 4620 × cxp 5664 ran crn 5667 “ cima 5669 ⟶wf 6529 –1-1-onto→wf1o 6532 1oc1o 8454 inlcinl 9890 inrcinr 9891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-om 7849 df-1st 7968 df-2nd 7969 df-1o 8461 df-inl 9893 df-inr 9894 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |