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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 2668 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) → 𝐴 ∈ V) | |
2 | elex 2668 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | 2 | adantr 272 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ V) |
4 | eleq1 2177 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
5 | eleq1 2177 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
6 | 5 | notbid 639 | . . . 4 ⊢ (𝑥 = 𝐴 → (¬ 𝑥 ∈ 𝐶 ↔ ¬ 𝐴 ∈ 𝐶)) |
7 | 4, 6 | anbi12d 462 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))) |
8 | df-dif 3039 | . . 3 ⊢ (𝐵 ∖ 𝐶) = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐶)} | |
9 | 7, 8 | elab2g 2800 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶))) |
10 | 1, 3, 9 | pm5.21nii 676 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 103 ↔ wb 104 = wceq 1314 ∈ wcel 1463 Vcvv 2657 ∖ cdif 3034 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-dif 3039 |
This theorem is referenced by: eldifd 3047 eldifad 3048 eldifbd 3049 difeqri 3162 eldifi 3164 eldifn 3165 difdif 3167 ddifstab 3174 ssconb 3175 sscon 3176 ssdif 3177 raldifb 3182 dfss4st 3275 ssddif 3276 unssdif 3277 inssdif 3278 difin 3279 unssin 3281 inssun 3282 invdif 3284 indif 3285 difundi 3294 difindiss 3296 indifdir 3298 undif3ss 3303 difin2 3304 symdifxor 3308 dfnul2 3331 reldisj 3380 disj3 3381 undif4 3391 ssdif0im 3393 inssdif0im 3396 ssundifim 3412 eldifsn 3616 difprsnss 3624 iundif2ss 3844 iindif2m 3846 brdif 3943 unidif0 4051 eldifpw 4358 elirr 4416 en2lp 4429 difopab 4632 intirr 4883 cnvdif 4903 imadiflem 5160 imadif 5161 elfi2 6812 xrlenlt 7753 nzadd 9010 irradd 9340 irrmul 9341 fzdifsuc 9754 fisumss 11053 inffinp1 11787 |
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