Theorem en1eqsn 9185

Description: A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) Avoid ax-pow 5307, ax-un 7689. (Revised by BTernaryTau, 4-Jan-2025.)

Assertion

Ref	Expression
en1eqsn	⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴})

Proof of Theorem en1eqsn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		en1 8971	. . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑥 𝐵 = {𝑥})
2		eleq2 2825	. . . . . . . 8 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ {𝑥}))
3		elsni 4584	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
4	3	sneqd 4579	. . . . . . . 8 ⊢ (𝐴 ∈ {𝑥} → {𝐴} = {𝑥})
5	2, 4	biimtrdi 253	. . . . . . 7 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → {𝐴} = {𝑥}))
6	5	imp 406	. . . . . 6 ⊢ ((𝐵 = {𝑥} ∧ 𝐴 ∈ 𝐵) → {𝐴} = {𝑥})
7		eqtr3 2758	. . . . . 6 ⊢ ((𝐵 = {𝑥} ∧ {𝐴} = {𝑥}) → 𝐵 = {𝐴})
8	6, 7	syldan 592	. . . . 5 ⊢ ((𝐵 = {𝑥} ∧ 𝐴 ∈ 𝐵) → 𝐵 = {𝐴})
9	8	ex 412	. . . 4 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴}))
10	9	exlimiv 1932	. . 3 ⊢ (∃𝑥 𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴}))
11	1, 10	sylbi 217	. 2 ⊢ (𝐵 ≈ 1o → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴}))
12	11	impcom 407	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {csn 4567   class class class wbr 5085  1_oc1o 8398   ≈ cen 8890

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-1o 8405  df-en 8894

This theorem is referenced by:  en1eqsnbi  9186  1nsgtrivd  19149  gex1  19566  0cyg  19868  pgpfac1lem3a  20053  pgpfaclem3  20060  0ring  20503  en1top  22949  cnextfres1  24033  xrge0tsmseq  33136  sconnpi1  35421  rngoueqz  38261  isdmn3  38395