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Theorem en1b 9065
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7755. (Revised by BTernaryTau, 24-Sep-2024.)
Assertion
Ref Expression
en1b (𝐴 ≈ 1o𝐴 = { 𝐴})

Proof of Theorem en1b
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 en1 9064 . . 3 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
2 id 22 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
3 unieq 4918 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 unisnv 4927 . . . . . . 7 {𝑥} = 𝑥
53, 4eqtrdi 2793 . . . . . 6 (𝐴 = {𝑥} → 𝐴 = 𝑥)
65sneqd 4638 . . . . 5 (𝐴 = {𝑥} → { 𝐴} = {𝑥})
72, 6eqtr4d 2780 . . . 4 (𝐴 = {𝑥} → 𝐴 = { 𝐴})
87exlimiv 1930 . . 3 (∃𝑥 𝐴 = {𝑥} → 𝐴 = { 𝐴})
91, 8sylbi 217 . 2 (𝐴 ≈ 1o𝐴 = { 𝐴})
10 id 22 . . 3 (𝐴 = { 𝐴} → 𝐴 = { 𝐴})
11 eqsnuniex 5361 . . . 4 (𝐴 = { 𝐴} → 𝐴 ∈ V)
12 ensn1g 9062 . . . 4 ( 𝐴 ∈ V → { 𝐴} ≈ 1o)
1311, 12syl 17 . . 3 (𝐴 = { 𝐴} → { 𝐴} ≈ 1o)
1410, 13eqbrtrd 5165 . 2 (𝐴 = { 𝐴} → 𝐴 ≈ 1o)
159, 14impbii 209 1 (𝐴 ≈ 1o𝐴 = { 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  {csn 4626   cuni 4907   class class class wbr 5143  1oc1o 8499  cen 8982
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1o 8506  df-en 8986
This theorem is referenced by:  en1uniel  9069  sylow2alem2  19636  sylow2a  19637  frgpcyg  21592  ptcmplem3  24062  cnextfvval  24073  cnextcn  24075  minveclem4a  25464  isppw  27157  xrge0tsmsbi  33066
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