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Mirrors > Home > ILE Home > Th. List > elin | GIF version |
Description: Expansion of membership in an intersection of two classes. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 29-Apr-1994.) |
Ref | Expression |
---|---|
elin | ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2671 | . 2 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) → 𝐴 ∈ V) | |
2 | elex 2671 | . . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
3 | 2 | adantl 275 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶) → 𝐴 ∈ V) |
4 | eleq1 2180 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
5 | eleq1 2180 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
6 | 4, 5 | anbi12d 464 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))) |
7 | df-in 3047 | . . 3 ⊢ (𝐵 ∩ 𝐶) = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶)} | |
8 | 6, 7 | elab2g 2804 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))) |
9 | 1, 3, 8 | pm5.21nii 678 | 1 ⊢ (𝐴 ∈ (𝐵 ∩ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1316 ∈ wcel 1465 Vcvv 2660 ∩ cin 3040 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-v 2662 df-in 3047 |
This theorem is referenced by: elini 3230 elind 3231 elinel1 3232 elinel2 3233 elin2 3234 elin3 3237 incom 3238 ineqri 3239 ineq1 3240 inass 3256 inss1 3266 ssin 3268 ssrin 3271 dfss4st 3279 inssdif 3282 difin 3283 unssin 3285 inssun 3286 invdif 3288 indif 3289 indi 3293 undi 3294 difundi 3298 difindiss 3300 indifdir 3302 difin2 3308 inrab2 3319 inelcm 3393 inssdif0im 3400 uniin 3726 intun 3772 intpr 3773 elrint 3781 iunin2 3846 iinin2m 3851 elriin 3853 disjnim 3890 disjiun 3894 brin 3950 trin 4006 inex1 4032 inuni 4050 bnd2 4067 ordpwsucss 4452 ordpwsucexmid 4455 peano5 4482 inopab 4641 inxp 4643 dmin 4717 opelres 4794 intasym 4893 asymref 4894 dminss 4923 imainss 4924 inimasn 4926 ssrnres 4951 cnvresima 4998 dfco2a 5009 funinsn 5142 imainlem 5174 imain 5175 2elresin 5204 nfvres 5422 respreima 5516 isoini 5687 offval 5957 tfrlem5 6179 mapval2 6540 ixpin 6585 ssenen 6713 fnfi 6793 peano5nnnn 7668 peano5nni 8687 ixxdisj 9641 icodisj 9730 fzdisj 9787 uzdisj 9828 nn0disj 9870 fzouzdisj 9912 isumss 11115 fsumsplit 11131 sumsplitdc 11156 fsum2dlemstep 11158 isbasis2g 12123 tgval2 12131 tgcl 12144 epttop 12170 ssntr 12202 ntreq0 12212 cnptopresti 12318 cnptoprest 12319 cnptoprest2 12320 lmss 12326 txcnp 12351 txcnmpt 12353 bldisj 12481 blininf 12504 blres 12514 metrest 12586 pilem1 12771 bdinex1 12993 bj-indind 13026 |
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