Theorem en1b 8813

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7588. (Revised by BTernaryTau, 24-Sep-2024.)

Assertion

Ref	Expression
en1b	⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})

Proof of Theorem en1b

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		en1 8811	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
2		id 22	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥})
3		unieq 4850	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		vex 3436	. . . . . . . 8 ⊢ 𝑥 ∈ V
5	4	unisn 4861	. . . . . . 7 ⊢ ∪ {𝑥} = 𝑥
6	3, 5	eqtrdi 2794	. . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥)
7	6	sneqd 4573	. . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥})
8	2, 7	eqtr4d 2781	. . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
9	8	exlimiv 1933	. . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
10	1, 9	sylbi 216	. 2 ⊢ (𝐴 ≈ 1o → 𝐴 = {∪ 𝐴})
11		id 22	. . 3 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 = {∪ 𝐴})
12		eqsnuniex 5283	. . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V)
13		ensn1g 8809	. . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o)
14	12, 13	syl 17	. . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1o)
15	11, 14	eqbrtrd 5096	. 2 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1o)
16	10, 15	impbii 208	1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})

Syntax hints:   ↔ wb 205   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432  {csn 4561  ∪ cuni 4839   class class class wbr 5074  1_oc1o 8290   ≈ cen 8730

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-1o 8297  df-en 8734

This theorem is referenced by:  en1uniel  8818  en1unielOLD  8819  sylow2alem2  19223  sylow2a  19224  frgpcyg  20781  ptcmplem3  23205  cnextfvval  23216  cnextcn  23218  minveclem4a  24594  isppw  26263  xrge0tsmsbi  31318