Theorem en1b 8957

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7675. (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 8956	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
2		id 22	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥})
3		unieq 4872	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		unisnv 4881	. . . . . . 7 ⊢ ∪ {𝑥} = 𝑥
5	3, 4	eqtrdi 2780	. . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥)
6	5	sneqd 4591	. . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥})
7	2, 6	eqtr4d 2767	. . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴})
8	7	exlimiv 1930	. . 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 8954	. . . 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 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3438  {csn 4579  ∪ cuni 4861   class class class wbr 5095  1_oc1o 8388   ≈ cen 8876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-1o 8395  df-en 8880

This theorem is referenced by:  en1uniel  8961  sylow2alem2  19515  sylow2a  19516  frgpcyg  21498  ptcmplem3  23957  cnextfvval  23968  cnextcn  23970  minveclem4a  25346  isppw  27040  xrge0tsmsbi  33029