Theorem en1eqsn 9277

Description: A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) Avoid ax-pow 5330, ax-un 7758. (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 9062	. . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑥 𝐵 = {𝑥})
2		eleq2 2852	. . . . . . . 8 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ {𝑥}))
3		elsni 4600	. . . . . . . . 9 ⊢ (𝐴 ∈ {𝑥} → 𝐴 = 𝑥)
4	3	sneqd 4595	. . . . . . . 8 ⊢ (𝐴 ∈ {𝑥} → {𝐴} = {𝑥})
5	2, 4	biimtrdi 256	. . . . . . 7 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → {𝐴} = {𝑥}))
6	5	imp 412	. . . . . 6 ⊢ ((𝐵 = {𝑥} ∧ 𝐴 ∈ 𝐵) → {𝐴} = {𝑥})
7		eqtr3 2785	. . . . . 6 ⊢ ((𝐵 = {𝑥} ∧ {𝐴} = {𝑥}) → 𝐵 = {𝐴})
8	6, 7	syldan 603	. . . . 5 ⊢ ((𝐵 = {𝑥} ∧ 𝐴 ∈ 𝐵) → 𝐵 = {𝐴})
9	8	ex 418	. . . 4 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴}))
10	9	exlimiv 1964	. . 3 ⊢ (∃𝑥 𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴}))
11	1, 10	sylbi 220	. 2 ⊢ (𝐵 ≈ 1o → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴}))
12	11	impcom 413	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2146  {csn 4587   class class class wbr 5105  1_oc1o 8480   ≈ cen 8981

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-1o 8487  df-en 8985

This theorem is used by:  en1eqsnbi  9278  1nsgtrivd  19396  gex1  19817  0cyg  20119  pgpfac1lem3a  20304  pgpfaclem3  20311  0ring  20789  en1top  23314  cnextfres1  24399  xrge0tsmseq  33689  sconnpi1  36125  rngoueqz  39014  isdmn3  39148  isidom3  49581