Theorem en1b 8958

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7677. (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 8957	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
2		id 22	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥})
3		unieq 4871	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		unisnv 4880	. . . . . . 7 ⊢ ∪ {𝑥} = 𝑥
5	3, 4	eqtrdi 2784	. . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥)
6	5	sneqd 4589	. . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥})
7	2, 6	eqtr4d 2771	. . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
8	7	exlimiv 1931	. . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
9	1, 8	sylbi 217	. 2 ⊢ (𝐴 ≈ 1o → 𝐴 = {∪ 𝐴})
10		id 22	. . 3 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 = {∪ 𝐴})
11		eqsnuniex 5303	. . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V)
12		ensn1g 8955	. . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o)
13	11, 12	syl 17	. . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1o)
14	10, 13	eqbrtrd 5117	. 2 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1o)
15	9, 14	impbii 209	1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})

Syntax hints:   ↔ wb 206   = wceq 1541  ∃wex 1780   ∈ wcel 2113  Vcvv 3437  {csn 4577  ∪ cuni 4860   class class class wbr 5095  1_oc1o 8387   ≈ cen 8876

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 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-1o 8394  df-en 8880

This theorem is referenced by:  en1uniel  8962  sylow2alem2  19538  sylow2a  19539  frgpcyg  21519  ptcmplem3  23989  cnextfvval  24000  cnextcn  24002  minveclem4a  25377  isppw  27071  xrge0tsmsbi  33084