![]() |
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 7748. (Revised by BTernaryTau, 24-Sep-2024.) |
Ref | Expression |
---|---|
en1uniel | ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en1b 9056 | . . . 4 ⊢ (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆}) | |
2 | eqsnuniex 5365 | . . . 4 ⊢ (𝑆 = {∪ 𝑆} → ∪ 𝑆 ∈ V) | |
3 | 1, 2 | sylbi 216 | . . 3 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ V) |
4 | snidg 4667 | . . 3 ⊢ (∪ 𝑆 ∈ V → ∪ 𝑆 ∈ {∪ 𝑆}) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ {∪ 𝑆}) |
6 | 1 | biimpi 215 | . 2 ⊢ (𝑆 ≈ 1o → 𝑆 = {∪ 𝑆}) |
7 | 5, 6 | eleqtrrd 2832 | 1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 {csn 4632 ∪ cuni 4912 class class class wbr 5152 1oc1o 8488 ≈ cen 8969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-1o 8495 df-en 8973 |
This theorem is referenced by: en2eleq 10041 en2other2 10042 pmtrf 19424 pmtrmvd 19425 pmtrfinv 19430 frgpcyg 21521 |
Copyright terms: Public domain | W3C validator |