| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > en1uniel | Structured version Visualization version GIF version | ||
| Description: A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7680. (Revised by BTernaryTau, 24-Sep-2024.) |
| Ref | Expression |
|---|---|
| en1uniel | ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | en1b 8962 | . . . 4 ⊢ (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆}) | |
| 2 | eqsnuniex 5306 | . . . 4 ⊢ (𝑆 = {∪ 𝑆} → ∪ 𝑆 ∈ V) | |
| 3 | 1, 2 | sylbi 217 | . . 3 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ V) |
| 4 | snidg 4617 | . . 3 ⊢ (∪ 𝑆 ∈ V → ∪ 𝑆 ∈ {∪ 𝑆}) | |
| 5 | 3, 4 | syl 17 | . 2 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ {∪ 𝑆}) |
| 6 | 1 | biimpi 216 | . 2 ⊢ (𝑆 ≈ 1o → 𝑆 = {∪ 𝑆}) |
| 7 | 5, 6 | eleqtrrd 2839 | 1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 Vcvv 3440 {csn 4580 ∪ cuni 4863 class class class wbr 5098 1oc1o 8390 ≈ cen 8880 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-1o 8397 df-en 8884 |
| This theorem is referenced by: en2eleq 9918 en2other2 9919 pmtrf 19384 pmtrmvd 19385 pmtrfinv 19390 frgpcyg 21528 |
| Copyright terms: Public domain | W3C validator |