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Theorem en1b 7968
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 7967 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
2 id 22 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
3 unieq 4410 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 vex 3189 . . . . . . . 8 𝑥 ∈ V
54unisn 4417 . . . . . . 7 {𝑥} = 𝑥
63, 5syl6eq 2671 . . . . . 6 (𝐴 = {𝑥} → 𝐴 = 𝑥)
76sneqd 4160 . . . . 5 (𝐴 = {𝑥} → { 𝐴} = {𝑥})
82, 7eqtr4d 2658 . . . 4 (𝐴 = {𝑥} → 𝐴 = { 𝐴})
98exlimiv 1855 . . 3 (∃𝑥 𝐴 = {𝑥} → 𝐴 = { 𝐴})
101, 9sylbi 207 . 2 (𝐴 ≈ 1𝑜𝐴 = { 𝐴})
11 id 22 . . 3 (𝐴 = { 𝐴} → 𝐴 = { 𝐴})
12 snex 4869 . . . . . 6 { 𝐴} ∈ V
1311, 12syl6eqel 2706 . . . . 5 (𝐴 = { 𝐴} → 𝐴 ∈ V)
14 uniexg 6908 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ V)
1513, 14syl 17 . . . 4 (𝐴 = { 𝐴} → 𝐴 ∈ V)
16 ensn1g 7965 . . . 4 ( 𝐴 ∈ V → { 𝐴} ≈ 1𝑜)
1715, 16syl 17 . . 3 (𝐴 = { 𝐴} → { 𝐴} ≈ 1𝑜)
1811, 17eqbrtrd 4635 . 2 (𝐴 = { 𝐴} → 𝐴 ≈ 1𝑜)
1910, 18impbii 199 1 (𝐴 ≈ 1𝑜𝐴 = { 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  {csn 4148   cuni 4402   class class class wbr 4613  1𝑜c1o 7498  cen 7896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-1o 7505  df-en 7900
This theorem is referenced by:  en1uniel  7972  sylow2alem2  17954  sylow2a  17955  frgpcyg  19841  ptcmplem3  21768  cnextfvval  21779  cnextcn  21781  minveclem4a  23109  isppw  24740  xrge0tsmsbi  29571
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