Theorem en1b 7910

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 7909	. . 3 ⊢ (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
2		id 22	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥})
3		unieq 4380	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		vex 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
5	4	unisn 4387	. . . . . . 7 ⊢ ∪ {𝑥} = 𝑥
6	3, 5	syl6eq 2660	. . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥)
7	6	sneqd 4137	. . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥})
8	2, 7	eqtr4d 2647	. . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
9	8	exlimiv 1845	. . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
10	1, 9	sylbi 206	. 2 ⊢ (𝐴 ≈ 1𝑜 → 𝐴 = {∪ 𝐴})
11		id 22	. . 3 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 = {∪ 𝐴})
12		snex 4835	. . . . . 6 ⊢ {∪ 𝐴} ∈ V
13	11, 12	syl6eqel 2696	. . . . 5 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ∈ V)
14		uniexg 6853	. . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
15	13, 14	syl 17	. . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V)
16		ensn1g 7907	. . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1𝑜)
17	15, 16	syl 17	. . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1𝑜)
18	11, 17	eqbrtrd 4605	. 2 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1𝑜)
19	10, 18	impbii 198	1 ⊢ (𝐴 ≈ 1𝑜 ↔ 𝐴 = {∪ 𝐴})

Syntax hints:   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  {csn 4125  ∪ cuni 4372   class class class wbr 4583  1_𝑜c1o 7440   ≈ cen 7838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-en 7842

This theorem is referenced by:  en1uniel  7914  sylow2alem2  17856  sylow2a  17857  frgpcyg  19741  ptcmplem3  21668  cnextfvval  21679  cnextcn  21681  minveclem4a  23009  isppw  24640  xrge0tsmsbi  29117