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Theorem en1b 8177
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 8176 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
2 id 22 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
3 unieq 4582 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 vex 3354 . . . . . . . 8 𝑥 ∈ V
54unisn 4589 . . . . . . 7 {𝑥} = 𝑥
63, 5syl6eq 2821 . . . . . 6 (𝐴 = {𝑥} → 𝐴 = 𝑥)
76sneqd 4328 . . . . 5 (𝐴 = {𝑥} → { 𝐴} = {𝑥})
82, 7eqtr4d 2808 . . . 4 (𝐴 = {𝑥} → 𝐴 = { 𝐴})
98exlimiv 2010 . . 3 (∃𝑥 𝐴 = {𝑥} → 𝐴 = { 𝐴})
101, 9sylbi 207 . 2 (𝐴 ≈ 1𝑜𝐴 = { 𝐴})
11 id 22 . . 3 (𝐴 = { 𝐴} → 𝐴 = { 𝐴})
12 snex 5036 . . . . . 6 { 𝐴} ∈ V
1311, 12syl6eqel 2858 . . . . 5 (𝐴 = { 𝐴} → 𝐴 ∈ V)
14 uniexg 7102 . . . . 5 (𝐴 ∈ V → 𝐴 ∈ V)
1513, 14syl 17 . . . 4 (𝐴 = { 𝐴} → 𝐴 ∈ V)
16 ensn1g 8174 . . . 4 ( 𝐴 ∈ V → { 𝐴} ≈ 1𝑜)
1715, 16syl 17 . . 3 (𝐴 = { 𝐴} → { 𝐴} ≈ 1𝑜)
1811, 17eqbrtrd 4808 . 2 (𝐴 = { 𝐴} → 𝐴 ≈ 1𝑜)
1910, 18impbii 199 1 (𝐴 ≈ 1𝑜𝐴 = { 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  {csn 4316   cuni 4574   class class class wbr 4786  1𝑜c1o 7706  cen 8106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1o 7713  df-en 8110
This theorem is referenced by:  en1uniel  8181  sylow2alem2  18240  sylow2a  18241  frgpcyg  20137  ptcmplem3  22078  cnextfvval  22089  cnextcn  22091  minveclem4a  23420  isppw  25061  xrge0tsmsbi  30126
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