Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  en1b Structured version   Visualization version   GIF version

Theorem en1b 8609
 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 8608 . . 3 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
2 id 22 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
3 unieq 4812 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 = {𝑥})
4 vex 3413 . . . . . . . 8 𝑥 ∈ V
54unisn 4823 . . . . . . 7 {𝑥} = 𝑥
63, 5eqtrdi 2809 . . . . . 6 (𝐴 = {𝑥} → 𝐴 = 𝑥)
76sneqd 4537 . . . . 5 (𝐴 = {𝑥} → { 𝐴} = {𝑥})
82, 7eqtr4d 2796 . . . 4 (𝐴 = {𝑥} → 𝐴 = { 𝐴})
98exlimiv 1931 . . 3 (∃𝑥 𝐴 = {𝑥} → 𝐴 = { 𝐴})
101, 9sylbi 220 . 2 (𝐴 ≈ 1o𝐴 = { 𝐴})
11 id 22 . . 3 (𝐴 = { 𝐴} → 𝐴 = { 𝐴})
12 snex 5304 . . . . . 6 { 𝐴} ∈ V
1311, 12eqeltrdi 2860 . . . . 5 (𝐴 = { 𝐴} → 𝐴 ∈ V)
1413uniexd 7472 . . . 4 (𝐴 = { 𝐴} → 𝐴 ∈ V)
15 ensn1g 8606 . . . 4 ( 𝐴 ∈ V → { 𝐴} ≈ 1o)
1614, 15syl 17 . . 3 (𝐴 = { 𝐴} → { 𝐴} ≈ 1o)
1711, 16eqbrtrd 5058 . 2 (𝐴 = { 𝐴} → 𝐴 ≈ 1o)
1810, 17impbii 212 1 (𝐴 ≈ 1o𝐴 = { 𝐴})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3409  {csn 4525  ∪ cuni 4801   class class class wbr 5036  1oc1o 8111   ≈ cen 8537 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-1o 8118  df-en 8541 This theorem is referenced by:  en1uniel  8613  sylow2alem2  18824  sylow2a  18825  frgpcyg  20355  ptcmplem3  22768  cnextfvval  22779  cnextcn  22781  minveclem4a  24144  isppw  25812  xrge0tsmsbi  30857
 Copyright terms: Public domain W3C validator