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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 6646 | . . . 4 ⊢ Rel ≈ | |
2 | 1 | brrelex1i 4590 | . . 3 ⊢ (𝑆 ≈ 1o → 𝑆 ∈ V) |
3 | uniexg 4369 | . . 3 ⊢ (𝑆 ∈ V → ∪ 𝑆 ∈ V) | |
4 | snidg 3561 | . . 3 ⊢ (∪ 𝑆 ∈ V → ∪ 𝑆 ∈ {∪ 𝑆}) | |
5 | 2, 3, 4 | 3syl 17 | . 2 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ {∪ 𝑆}) |
6 | encv 6648 | . . . . 5 ⊢ (𝑆 ≈ 1o → (𝑆 ∈ V ∧ 1o ∈ V)) | |
7 | 6 | simpld 111 | . . . 4 ⊢ (𝑆 ≈ 1o → 𝑆 ∈ V) |
8 | en1bg 6702 | . . . 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 2220 | 1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1332 ∈ wcel 1481 Vcvv 2689 {csn 3532 ∪ cuni 3744 class class class wbr 3937 1oc1o 6314 ≈ cen 6640 |
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 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-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-1o 6321 df-en 6643 |
This theorem is referenced by: en2eleq 7068 en2other2 7069 |
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