Theorem en1eqsn 9273

Description: A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) Avoid ax-pow 5363, ax-un 7724. (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 9020	. . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑥 𝐵 = {𝑥})
2		eleq2 2822	. . . . . . . 8 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ {𝑥}))
3		elsni 4645	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
4	3	sneqd 4640	. . . . . . . 8 ⊢ (𝐴 ∈ {𝑥} → {𝐴} = {𝑥})
5	2, 4	syl6bi 252	. . . . . . 7 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → {𝐴} = {𝑥}))
6	5	imp 407	. . . . . 6 ⊢ ((𝐵 = {𝑥} ∧ 𝐴 ∈ 𝐵) → {𝐴} = {𝑥})
7		eqtr3 2758	. . . . . 6 ⊢ ((𝐵 = {𝑥} ∧ {𝐴} = {𝑥}) → 𝐵 = {𝐴})
8	6, 7	syldan 591	. . . . 5 ⊢ ((𝐵 = {𝑥} ∧ 𝐴 ∈ 𝐵) → 𝐵 = {𝐴})
9	8	ex 413	. . . 4 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴}))
10	9	exlimiv 1933	. . 3 ⊢ (∃𝑥 𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴}))
11	1, 10	sylbi 216	. 2 ⊢ (𝐵 ≈ 1o → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴}))
12	11	impcom 408	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {csn 4628   class class class wbr 5148  1_oc1o 8458   ≈ cen 8935

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-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1o 8465  df-en 8939

This theorem is referenced by:  en1eqsnbi  9275  1nsgtrivd  19053  gex1  19458  0cyg  19760  pgpfac1lem3a  19945  pgpfaclem3  19952  0ring  20302  en1top  22486  cnextfres1  23571  xrge0tsmseq  32206  sconnpi1  34225  rngoueqz  36803  isdmn3  36937