ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sssnm GIF version

Theorem sssnm 3681
Description: The inhabited subset of a singleton. (Contributed by Jim Kingdon, 10-Aug-2018.)
Assertion
Ref Expression
sssnm (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem sssnm
StepHypRef Expression
1 ssel 3091 . . . . . . . . . 10 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 ∈ {𝐵}))
2 elsni 3545 . . . . . . . . . 10 (𝑥 ∈ {𝐵} → 𝑥 = 𝐵)
31, 2syl6 33 . . . . . . . . 9 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝑥 = 𝐵))
4 eleq1 2202 . . . . . . . . 9 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
53, 4syl6 33 . . . . . . . 8 (𝐴 ⊆ {𝐵} → (𝑥𝐴 → (𝑥𝐴𝐵𝐴)))
65ibd 177 . . . . . . 7 (𝐴 ⊆ {𝐵} → (𝑥𝐴𝐵𝐴))
76exlimdv 1791 . . . . . 6 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴𝐵𝐴))
8 snssi 3664 . . . . . 6 (𝐵𝐴 → {𝐵} ⊆ 𝐴)
97, 8syl6 33 . . . . 5 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴 → {𝐵} ⊆ 𝐴))
109anc2li 327 . . . 4 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴)))
11 eqss 3112 . . . 4 (𝐴 = {𝐵} ↔ (𝐴 ⊆ {𝐵} ∧ {𝐵} ⊆ 𝐴))
1210, 11syl6ibr 161 . . 3 (𝐴 ⊆ {𝐵} → (∃𝑥 𝑥𝐴𝐴 = {𝐵}))
1312com12 30 . 2 (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} → 𝐴 = {𝐵}))
14 eqimss 3151 . 2 (𝐴 = {𝐵} → 𝐴 ⊆ {𝐵})
1513, 14impbid1 141 1 (∃𝑥 𝑥𝐴 → (𝐴 ⊆ {𝐵} ↔ 𝐴 = {𝐵}))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480  wss 3071  {csn 3527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-sn 3533
This theorem is referenced by:  eqsnm  3682  exmid01  4121  exmidn0m  4124  exmidsssn  4125  exmidomni  7014  exmidunben  11950  exmidsbthrlem  13278  sbthomlem  13281
  Copyright terms: Public domain W3C validator