Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > incom | GIF version |
Description: Commutative law for intersection of classes. Exercise 7 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
incom | ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 264 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
2 | elin 3300 | . . 3 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
3 | elin 3300 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∩ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
4 | 1, 2, 3 | 3bitr4i 211 | . 2 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ 𝑥 ∈ (𝐵 ∩ 𝐴)) |
5 | 4 | eqriv 2161 | 1 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1342 ∈ wcel 2135 ∩ cin 3110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-in 3117 |
This theorem is referenced by: ineq2 3312 dfss1 3321 in12 3328 in32 3329 in13 3330 in31 3331 inss2 3338 sslin 3343 inss 3347 indif1 3362 indifcom 3363 indir 3366 symdif1 3382 dfrab2 3392 0in 3439 disjr 3453 ssdifin0 3485 difdifdirss 3488 uneqdifeqim 3489 diftpsn3 3708 iunin1 3924 iinin1m 3929 riinm 3932 rintm 3952 inex2 4111 onintexmid 4544 resiun1 4897 dmres 4899 rescom 4903 resima2 4912 xpssres 4913 resindm 4920 resdmdfsn 4921 resopab 4922 imadisj 4960 ndmima 4975 intirr 4984 djudisj 5025 imainrect 5043 dmresv 5056 resdmres 5089 funimaexg 5266 fnresdisj 5292 fnimaeq0 5303 resasplitss 5361 f0rn0 5376 fvun2 5547 ressnop0 5660 fvsnun1 5676 fsnunfv 5680 offres 6095 smores3 6252 phplem2 6810 unfiin 6882 xpfi 6886 endjusym 7052 djucomen 7163 fzpreddisj 9996 fseq1p1m1 10019 hashunlem 10706 zfz1isolem1 10739 fprodsplit 11524 znnen 12274 setsfun 12372 setsfun0 12373 setsslid 12387 restin 12723 metreslem 12927 bdinex2 13623 |
Copyright terms: Public domain | W3C validator |