Theorem en1uniel 8960

Description: A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7676. (Revised by BTernaryTau, 24-Sep-2024.)

Assertion

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

Proof of Theorem en1uniel

Step	Hyp	Ref	Expression
1		en1b 8956	. . . 4 ⊢ (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆})
2		eqsnuniex 5303	. . . 4 ⊢ (𝑆 = {∪ 𝑆} → ∪ 𝑆 ∈ V)
3	1, 2	sylbi 217	. . 3 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ V)
4		snidg 4614	. . 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 2836	1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3437  {csn 4577  ∪ cuni 4860   class class class wbr 5095  1_oc1o 8386   ≈ cen 8874

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 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-1o 8393  df-en 8878

This theorem is referenced by:  en2eleq  9908  en2other2  9909  pmtrf  19371  pmtrmvd  19372  pmtrfinv  19377  frgpcyg  21514