Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6631 | . . . 4 ⊢ Rel ≈ | |
2 | 1 | brrelex1i 4577 | . . 3 ⊢ (𝑆 ≈ 1o → 𝑆 ∈ V) |
3 | uniexg 4356 | . . 3 ⊢ (𝑆 ∈ V → ∪ 𝑆 ∈ V) | |
4 | snidg 3549 | . . 3 ⊢ (∪ 𝑆 ∈ V → ∪ 𝑆 ∈ {∪ 𝑆}) | |
5 | 2, 3, 4 | 3syl 17 | . 2 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ {∪ 𝑆}) |
6 | encv 6633 | . . . . 5 ⊢ (𝑆 ≈ 1o → (𝑆 ∈ V ∧ 1o ∈ V)) | |
7 | 6 | simpld 111 | . . . 4 ⊢ (𝑆 ≈ 1o → 𝑆 ∈ V) |
8 | en1bg 6687 | . . . 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 2217 | 1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2681 {csn 3522 ∪ cuni 3731 class class class wbr 3924 1oc1o 6299 ≈ cen 6625 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-1o 6306 df-en 6628 |
This theorem is referenced by: en2eleq 7044 en2other2 7045 |
Copyright terms: Public domain | W3C validator |