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Theorem en1b 8065
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 ≈ 1𝑜𝐴 = { 𝐴})

Proof of Theorem en1b
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 en1 8064 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
2 id 22 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
3 unieq 4476 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 vex 3234 . . . . . . . 8 𝑥 ∈ V
54unisn 4483 . . . . . . 7 {𝑥} = 𝑥
63, 5syl6eq 2701 . . . . . 6 (𝐴 = {𝑥} → 𝐴 = 𝑥)
76sneqd 4222 . . . . 5 (𝐴 = {𝑥} → { 𝐴} = {𝑥})
82, 7eqtr4d 2688 . . . 4 (𝐴 = {𝑥} → 𝐴 = { 𝐴})
98exlimiv 1898 . . 3 (∃𝑥 𝐴 = {𝑥} → 𝐴 = { 𝐴})
101, 9sylbi 207 . 2 (𝐴 ≈ 1𝑜𝐴 = { 𝐴})
11 id 22 . . 3 (𝐴 = { 𝐴} → 𝐴 = { 𝐴})
12 snex 4938 . . . . . 6 { 𝐴} ∈ V
1311, 12syl6eqel 2738 . . . . 5 (𝐴 = { 𝐴} → 𝐴 ∈ V)
14 uniexg 6997 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ V)
1513, 14syl 17 . . . 4 (𝐴 = { 𝐴} → 𝐴 ∈ V)
16 ensn1g 8062 . . . 4 ( 𝐴 ∈ V → { 𝐴} ≈ 1𝑜)
1715, 16syl 17 . . 3 (𝐴 = { 𝐴} → { 𝐴} ≈ 1𝑜)
1811, 17eqbrtrd 4707 . 2 (𝐴 = { 𝐴} → 𝐴 ≈ 1𝑜)
1910, 18impbii 199 1 (𝐴 ≈ 1𝑜𝐴 = { 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  {csn 4210   cuni 4468   class class class wbr 4685  1𝑜c1o 7598  cen 7994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1o 7605  df-en 7998
This theorem is referenced by:  en1uniel  8069  sylow2alem2  18079  sylow2a  18080  frgpcyg  19970  ptcmplem3  21905  cnextfvval  21916  cnextcn  21918  minveclem4a  23247  isppw  24885  xrge0tsmsbi  29914
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