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Mirrors > Home > ILE Home > Th. List > eldif | GIF version |
Description: Expansion of membership in a class difference. (Contributed by NM, 29-Apr-1994.) |
Ref | Expression |
---|---|
eldif | ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2723 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ V) | |
2 | elex 2723 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | 2 | adantr 274 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ V) |
4 | eleq1 2220 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
5 | eleq1 2220 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
6 | 5 | notbid 657 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶)) |
7 | 4, 6 | anbi12d 465 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))) |
8 | df-dif 3104 | . . 3 ⊢ (𝐵 ∖ 𝐶) = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)} | |
9 | 7, 8 | elab2g 2859 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))) |
10 | 1, 3, 9 | pm5.21nii 694 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 = wceq 1335 ∈ wcel 2128 Vcvv 2712 ∖ cdif 3099 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-tru 1338 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-v 2714 df-dif 3104 |
This theorem is referenced by: eldifd 3112 eldifad 3113 eldifbd 3114 difeqri 3228 eldifi 3230 eldifn 3231 difdif 3233 ddifstab 3240 ssconb 3241 sscon 3242 ssdif 3243 raldifb 3248 dfss4st 3341 ssddif 3342 unssdif 3343 inssdif 3344 difin 3345 unssin 3347 inssun 3348 invdif 3350 indif 3351 difundi 3360 difindiss 3362 indifdir 3364 undif3ss 3369 difin2 3370 symdifxor 3374 dfnul2 3397 reldisj 3446 disj3 3447 undif4 3457 ssdif0im 3459 inssdif0im 3462 ssundifim 3478 eldifpr 3588 eldiftp 3607 eldifsn 3688 difprsnss 3696 iundif2ss 3916 iindif2m 3918 brdif 4019 unidif0 4130 eldifpw 4439 elirr 4502 en2lp 4515 difopab 4721 intirr 4974 cnvdif 4994 imadiflem 5251 imadif 5252 elfi2 6918 xrlenlt 7944 nzadd 9224 irradd 9561 irrmul 9562 fzdifsuc 9989 fisumss 11300 prodssdc 11497 fprodssdc 11498 inffinp1 12228 bj-charfunr 13456 |
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