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Theorem en1bg 6216
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Jim Kingdon, 13-Apr-2020.)
Assertion
Ref Expression
en1bg (A 𝑉 → (A ≈ 1𝑜A = { A}))

Proof of Theorem en1bg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 en1 6215 . . 3 (A ≈ 1𝑜x A = {x})
2 id 19 . . . . 5 (A = {x} → A = {x})
3 unieq 3580 . . . . . . 7 (A = {x} → A = {x})
4 vex 2554 . . . . . . . 8 x V
54unisn 3587 . . . . . . 7 {x} = x
63, 5syl6eq 2085 . . . . . 6 (A = {x} → A = x)
76sneqd 3380 . . . . 5 (A = {x} → { A} = {x})
82, 7eqtr4d 2072 . . . 4 (A = {x} → A = { A})
98exlimiv 1486 . . 3 (x A = {x} → A = { A})
101, 9sylbi 114 . 2 (A ≈ 1𝑜A = { A})
11 uniexg 4141 . . . 4 (A 𝑉 A V)
12 ensn1g 6213 . . . 4 ( A V → { A} ≈ 1𝑜)
1311, 12syl 14 . . 3 (A 𝑉 → { A} ≈ 1𝑜)
14 breq1 3758 . . 3 (A = { A} → (A ≈ 1𝑜 ↔ { A} ≈ 1𝑜))
1513, 14syl5ibrcom 146 . 2 (A 𝑉 → (A = { A} → A ≈ 1𝑜))
1610, 15impbid2 131 1 (A 𝑉 → (A ≈ 1𝑜A = { A}))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  {csn 3367   cuni 3571   class class class wbr 3755  1𝑜c1o 5933  cen 6155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-1o 5940  df-en 6158
This theorem is referenced by:  en1uniel  6220
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