Theorem en1b 8065

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)

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 8064	. . 3 ⊢ (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
2		id 22	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥})
3		unieq 4476	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		vex 3234	. . . . . . . 8 ⊢ 𝑥 ∈ V
5	4	unisn 4483	. . . . . . 7 ⊢ ∪ {𝑥} = 𝑥
6	3, 5	syl6eq 2701	. . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥)
7	6	sneqd 4222	. . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥})
8	2, 7	eqtr4d 2688	. . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
9	8	exlimiv 1898	. . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
10	1, 9	sylbi 207	. 2 ⊢ (𝐴 ≈ 1𝑜 → 𝐴 = {∪ 𝐴})
11		id 22	. . 3 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 = {∪ 𝐴})
12		snex 4938	. . . . . 6 ⊢ {∪ 𝐴} ∈ V
13	11, 12	syl6eqel 2738	. . . . 5 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ∈ V)
14		uniexg 6997	. . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
15	13, 14	syl 17	. . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V)
16		ensn1g 8062	. . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1𝑜)
17	15, 16	syl 17	. . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1𝑜)
18	11, 17	eqbrtrd 4707	. 2 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1𝑜)
19	10, 18	impbii 199	1 ⊢ (𝐴 ≈ 1𝑜 ↔ 𝐴 = {∪ 𝐴})

Syntax hints:   ↔ wb 196   = wceq 1523  ∃wex 1744   ∈ wcel 2030  Vcvv 3231  {csn 4210  ∪ cuni 4468   class class class wbr 4685  1_𝑜c1o 7598   ≈ cen 7994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1o 7605  df-en 7998

This theorem is referenced by:  en1uniel  8069  sylow2alem2  18079  sylow2a  18080  frgpcyg  19970  ptcmplem3  21905  cnextfvval  21916  cnextcn  21918  minveclem4a  23247  isppw  24885  xrge0tsmsbi  29914