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

Theorem mss 3989
Description: An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)
Assertion
Ref Expression
mss (∃𝑦 𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑧 𝑧𝑥))
Distinct variable groups:   𝑥,𝑦   𝑥,𝑧   𝑥,𝐴,𝑦
Allowed substitution hint:   𝐴(𝑧)

Proof of Theorem mss
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 vex 2577 . . . . 5 𝑦 ∈ V
21snss 3521 . . . 4 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
31snm 3515 . . . . 5 𝑤 𝑤 ∈ {𝑦}
4 snexgOLD 3962 . . . . . . 7 (𝑦 ∈ V → {𝑦} ∈ V)
51, 4ax-mp 7 . . . . . 6 {𝑦} ∈ V
6 sseq1 2993 . . . . . . 7 (𝑥 = {𝑦} → (𝑥𝐴 ↔ {𝑦} ⊆ 𝐴))
7 eleq2 2117 . . . . . . . 8 (𝑥 = {𝑦} → (𝑤𝑥𝑤 ∈ {𝑦}))
87exbidv 1722 . . . . . . 7 (𝑥 = {𝑦} → (∃𝑤 𝑤𝑥 ↔ ∃𝑤 𝑤 ∈ {𝑦}))
96, 8anbi12d 450 . . . . . 6 (𝑥 = {𝑦} → ((𝑥𝐴 ∧ ∃𝑤 𝑤𝑥) ↔ ({𝑦} ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ {𝑦})))
105, 9spcev 2664 . . . . 5 (({𝑦} ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ {𝑦}) → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
113, 10mpan2 409 . . . 4 ({𝑦} ⊆ 𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
122, 11sylbi 118 . . 3 (𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
1312exlimiv 1505 . 2 (∃𝑦 𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
14 elequ1 1616 . . . . 5 (𝑧 = 𝑤 → (𝑧𝑥𝑤𝑥))
1514cbvexv 1811 . . . 4 (∃𝑧 𝑧𝑥 ↔ ∃𝑤 𝑤𝑥)
1615anbi2i 438 . . 3 ((𝑥𝐴 ∧ ∃𝑧 𝑧𝑥) ↔ (𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
1716exbii 1512 . 2 (∃𝑥(𝑥𝐴 ∧ ∃𝑧 𝑧𝑥) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
1813, 17sylibr 141 1 (∃𝑦 𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑧 𝑧𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574  wss 2944  {csn 3402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator