Theorem en1eqsn 9159

Description: A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) Avoid ax-pow 5301, ax-un 7668. (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 8946	. . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑥 𝐵 = {𝑥})
2		eleq2 2820	. . . . . . . 8 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ {𝑥}))
3		elsni 4590	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
4	3	sneqd 4585	. . . . . . . 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 2753	. . . . . 6 ⊢ ((𝐵 = {𝑥} ∧ {𝐴} = {𝑥}) → 𝐵 = {𝐴})
8	6, 7	syldan 591	. . . . 5 ⊢ ((𝐵 = {𝑥} ∧ 𝐴 ∈ 𝐵) → 𝐵 = {𝐴})
9	8	ex 412	. . . 4 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴}))
10	9	exlimiv 1931	. . 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 1541  ∃wex 1780   ∈ wcel 2111  {csn 4573   class class class wbr 5089  1_oc1o 8378   ≈ cen 8866

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 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-1o 8385  df-en 8870

This theorem is referenced by:  en1eqsnbi  9160  1nsgtrivd  19086  gex1  19503  0cyg  19805  pgpfac1lem3a  19990  pgpfaclem3  19997  0ring  20441  en1top  22899  cnextfres1  23983  xrge0tsmseq  33044  sconnpi1  35283  rngoueqz  37990  isdmn3  38124