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Theorem en1b 8974
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7677. (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 8972 . . 3 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
2 id 22 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
3 unieq 4881 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 unisnv 4893 . . . . . . 7 {𝑥} = 𝑥
53, 4eqtrdi 2793 . . . . . 6 (𝐴 = {𝑥} → 𝐴 = 𝑥)
65sneqd 4603 . . . . 5 (𝐴 = {𝑥} → { 𝐴} = {𝑥})
72, 6eqtr4d 2780 . . . 4 (𝐴 = {𝑥} → 𝐴 = { 𝐴})
87exlimiv 1934 . . 3 (∃𝑥 𝐴 = {𝑥} → 𝐴 = { 𝐴})
91, 8sylbi 216 . 2 (𝐴 ≈ 1o𝐴 = { 𝐴})
10 id 22 . . 3 (𝐴 = { 𝐴} → 𝐴 = { 𝐴})
11 eqsnuniex 5321 . . . 4 (𝐴 = { 𝐴} → 𝐴 ∈ V)
12 ensn1g 8970 . . . 4 ( 𝐴 ∈ V → { 𝐴} ≈ 1o)
1311, 12syl 17 . . 3 (𝐴 = { 𝐴} → { 𝐴} ≈ 1o)
1410, 13eqbrtrd 5132 . 2 (𝐴 = { 𝐴} → 𝐴 ≈ 1o)
159, 14impbii 208 1 (𝐴 ≈ 1o𝐴 = { 𝐴})
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wex 1782  wcel 2107  Vcvv 3448  {csn 4591   cuni 4870   class class class wbr 5110  1oc1o 8410  cen 8887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-1o 8417  df-en 8891
This theorem is referenced by:  en1uniel  8979  en1unielOLD  8980  sylow2alem2  19407  sylow2a  19408  frgpcyg  20996  ptcmplem3  23421  cnextfvval  23432  cnextcn  23434  minveclem4a  24810  isppw  26479  xrge0tsmsbi  31942
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