![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > in0 | Structured version Visualization version GIF version |
Description: The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 21-Jun-1993.) |
Ref | Expression |
---|---|
in0 | ⊢ (𝐴 ∩ ∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4344 | . . . 4 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | 1 | bianfi 533 | . . 3 ⊢ (𝑥 ∈ ∅ ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ∅)) |
3 | 2 | bicomi 224 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ ∅) ↔ 𝑥 ∈ ∅) |
4 | 3 | ineqri 4220 | 1 ⊢ (𝐴 ∩ ∅) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∩ cin 3962 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-dif 3966 df-in 3970 df-nul 4340 |
This theorem is referenced by: 0in 4403 csbin 4448 res0 6004 dfpo2 6318 predprc 6361 fresaun 6780 oev2 8560 dju0en 10214 ackbij1lem13 10269 ackbij1lem16 10272 incexclem 15869 bitsinv1 16476 bitsinvp1 16483 sadcadd 16492 sadadd2 16494 sadid1 16502 bitsres 16507 smumullem 16526 ressbas 17280 ressbasOLD 17281 sylow2a 19652 ablfac1eu 20108 indistopon 23024 fctop 23027 cctop 23029 rest0 23193 filconn 23907 volinun 25595 itg2cnlem2 25812 pthdlem2 29801 0pth 30154 1pthdlem2 30165 disjdifprg 32595 disjun0 32615 ofpreima2 32683 of0r 32695 ldgenpisyslem1 34144 0elcarsg 34289 carsgclctunlem1 34299 carsgclctunlem3 34302 ballotlemfval0 34477 sate0 35400 elima4 35757 bj-rest10 37071 bj-rest0 37076 mblfinlem2 37645 conrel1d 43653 conrel2d 43654 ntrk0kbimka 44029 clsneibex 44092 neicvgbex 44102 qinioo 45488 nnfoctbdjlem 46411 caragen0 46462 |
Copyright terms: Public domain | W3C validator |