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Theorem en1b 9064
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7754. (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 9063 . . 3 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
2 id 22 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
3 unieq 4923 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 unisnv 4932 . . . . . . 7 {𝑥} = 𝑥
53, 4eqtrdi 2791 . . . . . 6 (𝐴 = {𝑥} → 𝐴 = 𝑥)
65sneqd 4643 . . . . 5 (𝐴 = {𝑥} → { 𝐴} = {𝑥})
72, 6eqtr4d 2778 . . . 4 (𝐴 = {𝑥} → 𝐴 = { 𝐴})
87exlimiv 1928 . . 3 (∃𝑥 𝐴 = {𝑥} → 𝐴 = { 𝐴})
91, 8sylbi 217 . 2 (𝐴 ≈ 1o𝐴 = { 𝐴})
10 id 22 . . 3 (𝐴 = { 𝐴} → 𝐴 = { 𝐴})
11 eqsnuniex 5367 . . . 4 (𝐴 = { 𝐴} → 𝐴 ∈ V)
12 ensn1g 9061 . . . 4 ( 𝐴 ∈ V → { 𝐴} ≈ 1o)
1311, 12syl 17 . . 3 (𝐴 = { 𝐴} → { 𝐴} ≈ 1o)
1410, 13eqbrtrd 5170 . 2 (𝐴 = { 𝐴} → 𝐴 ≈ 1o)
159, 14impbii 209 1 (𝐴 ≈ 1o𝐴 = { 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  {csn 4631   cuni 4912   class class class wbr 5148  1oc1o 8498  cen 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1o 8505  df-en 8985
This theorem is referenced by:  en1uniel  9068  sylow2alem2  19651  sylow2a  19652  frgpcyg  21610  ptcmplem3  24078  cnextfvval  24089  cnextcn  24091  minveclem4a  25478  isppw  27172  xrge0tsmsbi  33049
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