Theorem en1b 8942

Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7663. (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 8941	. . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
2		id 22	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥})
3		unieq 4865	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥})
4		unisnv 4874	. . . . . . 7 ⊢ ∪ {𝑥} = 𝑥
5	3, 4	eqtrdi 2782	. . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥)
6	5	sneqd 4583	. . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥})
7	2, 6	eqtr4d 2769	. . . 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 5294	. . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V)
12		ensn1g 8939	. . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o)
13	11, 12	syl 17	. . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1o)
14	10, 13	eqbrtrd 5108	. 2 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1o)
15	9, 14	impbii 209	1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴})

Syntax hints:   ↔ wb 206   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436  {csn 4571  ∪ cuni 4854   class class class wbr 5086  1_oc1o 8373   ≈ cen 8861

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5229  ax-nul 5239  ax-pr 5365

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-br 5087  df-opab 5149  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-1o 8380  df-en 8865

This theorem is referenced by:  en1uniel  8946  sylow2alem2  19525  sylow2a  19526  frgpcyg  21505  ptcmplem3  23964  cnextfvval  23975  cnextcn  23977  minveclem4a  25352  isppw  27046  xrge0tsmsbi  33035