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Mirrors > Home > ILE Home > Th. List > elun | GIF version |
Description: Expansion of membership in class union. Theorem 12 of [Suppes] p. 25. (Contributed by NM, 7-Aug-1994.) |
Ref | Expression |
---|---|
elun | ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2700 | . 2 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → 𝐴 ∈ V) | |
2 | elex 2700 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | elex 2700 | . . 3 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
4 | 2, 3 | jaoi 706 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶) → 𝐴 ∈ V) |
5 | eleq1 2203 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
6 | eleq1 2203 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐶 ↔ 𝐴 ∈ 𝐶)) | |
7 | 5, 6 | orbi12d 783 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))) |
8 | df-un 3080 | . . 3 ⊢ (𝐵 ∪ 𝐶) = {𝑥 ∣ (𝑥 ∈ 𝐵 ∨ 𝑥 ∈ 𝐶)} | |
9 | 7, 8 | elab2g 2835 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))) |
10 | 1, 4, 9 | pm5.21nii 694 | 1 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶)) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∨ wo 698 = wceq 1332 ∈ wcel 1481 Vcvv 2689 ∪ cun 3074 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 |
This theorem is referenced by: uneqri 3223 uncom 3225 uneq1 3228 unass 3238 ssun1 3244 unss1 3250 ssequn1 3251 unss 3255 rexun 3261 ralunb 3262 unssdif 3316 unssin 3320 inssun 3321 indi 3328 undi 3329 difundi 3333 difindiss 3335 undif3ss 3342 symdifxor 3347 rabun2 3360 reuun2 3364 undif4 3430 ssundifim 3451 dcun 3478 dfpr2 3551 eltpg 3576 pwprss 3740 pwtpss 3741 uniun 3763 intun 3810 iunun 3899 iunxun 3900 iinuniss 3903 brun 3987 undifexmid 4125 exmidundif 4137 exmidundifim 4138 pwunss 4213 elsuci 4333 elsucg 4334 elsuc2g 4335 ordsucim 4424 sucprcreg 4472 opthprc 4598 xpundi 4603 xpundir 4604 funun 5175 mptun 5262 unpreima 5553 reldmtpos 6158 dftpos4 6168 tpostpos 6169 onunsnss 6813 unfidisj 6818 undifdcss 6819 fidcenumlemrks 6849 djulclb 6948 eldju 6961 eldju2ndl 6965 eldju2ndr 6966 ctssdccl 7004 elnn0 9003 un0addcl 9034 un0mulcl 9035 elxnn0 9066 ltxr 9592 elxr 9593 fzsplit2 9861 elfzp1 9883 uzsplit 9903 elfzp12 9910 fz01or 9922 fzosplit 9985 fzouzsplit 9987 elfzonlteqm1 10018 fzosplitsni 10043 hashinfuni 10555 hashennnuni 10557 hashunlem 10582 zfz1isolemiso 10614 summodclem3 11181 fsumsplit 11208 fsumsplitsn 11211 sumsplitdc 11233 reopnap 12746 djulclALT 13179 djurclALT 13180 bj-nntrans 13320 bj-nnelirr 13322 exmid1stab 13368 |
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