Theorem en1uniel 9048

Description: A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7734. (Revised by BTernaryTau, 24-Sep-2024.)

Assertion

Ref	Expression
en1uniel	⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)

Proof of Theorem en1uniel

Step	Hyp	Ref	Expression
1		en1b 9044	. . . 4 ⊢ (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆})
2		eqsnuniex 5336	. . . 4 ⊢ (𝑆 = {∪ 𝑆} → ∪ 𝑆 ∈ V)
3	1, 2	sylbi 217	. . 3 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ V)
4		snidg 4641	. . 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 2838	1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3464  {csn 4606  ∪ cuni 4888   class class class wbr 5124  1_oc1o 8478   ≈ cen 8961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1o 8485  df-en 8965

This theorem is referenced by:  en2eleq  10027  en2other2  10028  pmtrf  19441  pmtrmvd  19442  pmtrfinv  19447  frgpcyg  21539