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Theorem en1b 8569
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 8568 . . 3 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
2 id 22 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
3 unieq 4844 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 vex 3502 . . . . . . . 8 𝑥 ∈ V
54unisn 4852 . . . . . . 7 {𝑥} = 𝑥
63, 5syl6eq 2876 . . . . . 6 (𝐴 = {𝑥} → 𝐴 = 𝑥)
76sneqd 4575 . . . . 5 (𝐴 = {𝑥} → { 𝐴} = {𝑥})
82, 7eqtr4d 2863 . . . 4 (𝐴 = {𝑥} → 𝐴 = { 𝐴})
98exlimiv 1924 . . 3 (∃𝑥 𝐴 = {𝑥} → 𝐴 = { 𝐴})
101, 9sylbi 218 . 2 (𝐴 ≈ 1o𝐴 = { 𝐴})
11 id 22 . . 3 (𝐴 = { 𝐴} → 𝐴 = { 𝐴})
12 snex 5327 . . . . . 6 { 𝐴} ∈ V
1311, 12syl6eqel 2925 . . . . 5 (𝐴 = { 𝐴} → 𝐴 ∈ V)
14 uniexg 7460 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ V)
1513, 14syl 17 . . . 4 (𝐴 = { 𝐴} → 𝐴 ∈ V)
16 ensn1g 8566 . . . 4 ( 𝐴 ∈ V → { 𝐴} ≈ 1o)
1715, 16syl 17 . . 3 (𝐴 = { 𝐴} → { 𝐴} ≈ 1o)
1811, 17eqbrtrd 5084 . 2 (𝐴 = { 𝐴} → 𝐴 ≈ 1o)
1910, 18impbii 210 1 (𝐴 ≈ 1o𝐴 = { 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1530  wex 1773  wcel 2106  Vcvv 3499  {csn 4563   cuni 4836   class class class wbr 5062  1oc1o 8089  cen 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-1o 8096  df-en 8502
This theorem is referenced by:  en1uniel  8573  sylow2alem2  18665  sylow2a  18666  frgpcyg  20636  ptcmplem3  22578  cnextfvval  22589  cnextcn  22591  minveclem4a  23948  isppw  25605  xrge0tsmsbi  30608
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