Theorem en1b 8024

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 8023	. . 3 ⊢ (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
2		id 22	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥})
3		unieq 4444	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		vex 3203	. . . . . . . 8 ⊢ 𝑥 ∈ V
5	4	unisn 4451	. . . . . . 7 ⊢ ∪ {𝑥} = 𝑥
6	3, 5	syl6eq 2672	. . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥)
7	6	sneqd 4189	. . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥})
8	2, 7	eqtr4d 2659	. . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
9	8	exlimiv 1858	. . 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 4908	. . . . . 6 ⊢ {∪ 𝐴} ∈ V
13	11, 12	syl6eqel 2709	. . . . 5 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ∈ V)
14		uniexg 6955	. . . . 5 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
15	13, 14	syl 17	. . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V)
16		ensn1g 8021	. . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1𝑜)
17	15, 16	syl 17	. . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1𝑜)
18	11, 17	eqbrtrd 4675	. 2 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1𝑜)
19	10, 18	impbii 199	1 ⊢ (𝐴 ≈ 1𝑜 ↔ 𝐴 = {∪ 𝐴})

Syntax hints:   ↔ wb 196   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200  {csn 4177  ∪ cuni 4436   class class class wbr 4653  1_𝑜c1o 7553   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1o 7560  df-en 7956

This theorem is referenced by:  en1uniel  8028  sylow2alem2  18033  sylow2a  18034  frgpcyg  19922  ptcmplem3  21858  cnextfvval  21869  cnextcn  21871  minveclem4a  23201  isppw  24840  xrge0tsmsbi  29786