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Mirrors > Home > ILE Home > Th. List > en1uniel | GIF version |
Description: A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) |
Ref | Expression |
---|---|
en1uniel | ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relen 6606 | . . . 4 ⊢ Rel ≈ | |
2 | 1 | brrelex1i 4552 | . . 3 ⊢ (𝑆 ≈ 1o → 𝑆 ∈ V) |
3 | uniexg 4331 | . . 3 ⊢ (𝑆 ∈ V → ∪ 𝑆 ∈ V) | |
4 | snidg 3524 | . . 3 ⊢ (∪ 𝑆 ∈ V → ∪ 𝑆 ∈ {∪ 𝑆}) | |
5 | 2, 3, 4 | 3syl 17 | . 2 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ {∪ 𝑆}) |
6 | encv 6608 | . . . . 5 ⊢ (𝑆 ≈ 1o → (𝑆 ∈ V ∧ 1o ∈ V)) | |
7 | 6 | simpld 111 | . . . 4 ⊢ (𝑆 ≈ 1o → 𝑆 ∈ V) |
8 | en1bg 6662 | . . . 4 ⊢ (𝑆 ∈ V → (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆})) | |
9 | 7, 8 | syl 14 | . . 3 ⊢ (𝑆 ≈ 1o → (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆})) |
10 | 9 | ibi 175 | . 2 ⊢ (𝑆 ≈ 1o → 𝑆 = {∪ 𝑆}) |
11 | 5, 10 | eleqtrrd 2197 | 1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1316 ∈ wcel 1465 Vcvv 2660 {csn 3497 ∪ cuni 3706 class class class wbr 3899 1oc1o 6274 ≈ cen 6600 |
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 588 ax-in2 589 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-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-suc 4263 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-1o 6281 df-en 6603 |
This theorem is referenced by: en2eleq 7019 en2other2 7020 |
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