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 4180 | . 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ((inl “ 𝐴) ∩ (inr “ 𝐵)) | |
2 | imassrn 5942 | . . . 4 ⊢ (inr “ 𝐵) ⊆ ran inr | |
3 | djurf1o 9344 | . . . . 5 ⊢ inr:V–1-1-onto→({1o} × V) | |
4 | f1of 6617 | . . . . 5 ⊢ (inr:V–1-1-onto→({1o} × V) → inr:V⟶({1o} × V)) | |
5 | frn 6522 | . . . . 5 ⊢ (inr:V⟶({1o} × V) → ran inr ⊆ ({1o} × V)) | |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ ran inr ⊆ ({1o} × V) |
7 | 2, 6 | sstri 3978 | . . 3 ⊢ (inr “ 𝐵) ⊆ ({1o} × V) |
8 | incom 4180 | . . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = (({1o} × V) ∩ (inl “ 𝐴)) | |
9 | imassrn 5942 | . . . . . 6 ⊢ (inl “ 𝐴) ⊆ ran inl | |
10 | djulf1o 9343 | . . . . . . 7 ⊢ inl:V–1-1-onto→({∅} × V) | |
11 | f1of 6617 | . . . . . . 7 ⊢ (inl:V–1-1-onto→({∅} × V) → inl:V⟶({∅} × V)) | |
12 | frn 6522 | . . . . . . 7 ⊢ (inl:V⟶({∅} × V) → ran inl ⊆ ({∅} × V)) | |
13 | 10, 11, 12 | mp2b 10 | . . . . . 6 ⊢ ran inl ⊆ ({∅} × V) |
14 | 9, 13 | sstri 3978 | . . . . 5 ⊢ (inl “ 𝐴) ⊆ ({∅} × V) |
15 | 1n0 8121 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
16 | 15 | necomi 3072 | . . . . . 6 ⊢ ∅ ≠ 1o |
17 | disjsn2 4650 | . . . . . 6 ⊢ (∅ ≠ 1o → ({∅} ∩ {1o}) = ∅) | |
18 | xpdisj1 6020 | . . . . . 6 ⊢ (({∅} ∩ {1o}) = ∅ → (({∅} × V) ∩ ({1o} × V)) = ∅) | |
19 | 16, 17, 18 | mp2b 10 | . . . . 5 ⊢ (({∅} × V) ∩ ({1o} × V)) = ∅ |
20 | ssdisj 4411 | . . . . 5 ⊢ (((inl “ 𝐴) ⊆ ({∅} × V) ∧ (({∅} × V) ∩ ({1o} × V)) = ∅) → ((inl “ 𝐴) ∩ ({1o} × V)) = ∅) | |
21 | 14, 19, 20 | mp2an 690 | . . . 4 ⊢ ((inl “ 𝐴) ∩ ({1o} × V)) = ∅ |
22 | 8, 21 | eqtr3i 2848 | . . 3 ⊢ (({1o} × V) ∩ (inl “ 𝐴)) = ∅ |
23 | ssdisj 4411 | . . 3 ⊢ (((inr “ 𝐵) ⊆ ({1o} × V) ∧ (({1o} × V) ∩ (inl “ 𝐴)) = ∅) → ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅) | |
24 | 7, 22, 23 | mp2an 690 | . 2 ⊢ ((inr “ 𝐵) ∩ (inl “ 𝐴)) = ∅ |
25 | 1, 24 | eqtr3i 2848 | 1 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ≠ wne 3018 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 {csn 4569 × cxp 5555 ran crn 5558 “ cima 5560 ⟶wf 6353 –1-1-onto→wf1o 6356 1oc1o 8097 inlcinl 9330 inrcinr 9331 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-1st 7691 df-2nd 7692 df-1o 8104 df-inl 9333 df-inr 9334 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |