Theorem en1uniel 9024

Description: A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7721. (Revised by BTernaryTau, 24-Sep-2024.)

Assertion

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

Proof of Theorem en1uniel

Step	Hyp	Ref	Expression
1		en1b 9019	. . . 4 ⊢ (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆})
2		eqsnuniex 5358	. . . 4 ⊢ (𝑆 = {∪ 𝑆} → ∪ 𝑆 ∈ V)
3	1, 2	sylbi 216	. . 3 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ V)
4		snidg 4661	. . 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 2836	1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474  {csn 4627  ∪ cuni 4907   class class class wbr 5147  1_oc1o 8455   ≈ cen 8932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-1o 8462  df-en 8936

This theorem is referenced by:  en2eleq  9999  en2other2  10000  pmtrf  19317  pmtrmvd  19318  pmtrfinv  19323  frgpcyg  21120