Theorem en1uniel 9003

Description: A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.) Avoid ax-un 7714. (Revised by BTernaryTau, 24-Sep-2024.)

Assertion

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

Proof of Theorem en1uniel

Step	Hyp	Ref	Expression
1		en1b 8999	. . . 4 ⊢ (𝑆 ≈ 1o ↔ 𝑆 = {∪ 𝑆})
2		eqsnuniex 5319	. . . 4 ⊢ (𝑆 = {∪ 𝑆} → ∪ 𝑆 ∈ V)
3	1, 2	sylbi 217	. . 3 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ V)
4		snidg 4627	. . 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 2832	1 ⊢ (𝑆 ≈ 1o → ∪ 𝑆 ∈ 𝑆)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450  {csn 4592  ∪ cuni 4874   class class class wbr 5110  1_oc1o 8430   ≈ cen 8918

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-1o 8437  df-en 8922

This theorem is referenced by:  en2eleq  9968  en2other2  9969  pmtrf  19392  pmtrmvd  19393  pmtrfinv  19398  frgpcyg  21490