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Theorem mss 4017
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 2615 . . . . 5 𝑦 ∈ V
21snss 3540 . . . 4 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
31snm 3534 . . . . 5 𝑤 𝑤 ∈ {𝑦}
41snex 3984 . . . . . 6 {𝑦} ∈ V
5 sseq1 3031 . . . . . . 7 (𝑥 = {𝑦} → (𝑥𝐴 ↔ {𝑦} ⊆ 𝐴))
6 eleq2 2146 . . . . . . . 8 (𝑥 = {𝑦} → (𝑤𝑥𝑤 ∈ {𝑦}))
76exbidv 1748 . . . . . . 7 (𝑥 = {𝑦} → (∃𝑤 𝑤𝑥 ↔ ∃𝑤 𝑤 ∈ {𝑦}))
85, 7anbi12d 457 . . . . . 6 (𝑥 = {𝑦} → ((𝑥𝐴 ∧ ∃𝑤 𝑤𝑥) ↔ ({𝑦} ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ {𝑦})))
94, 8spcev 2703 . . . . 5 (({𝑦} ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ {𝑦}) → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
103, 9mpan2 416 . . . 4 ({𝑦} ⊆ 𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
112, 10sylbi 119 . . 3 (𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
1211exlimiv 1530 . 2 (∃𝑦 𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
13 elequ1 1642 . . . . 5 (𝑧 = 𝑤 → (𝑧𝑥𝑤𝑥))
1413cbvexv 1838 . . . 4 (∃𝑧 𝑧𝑥 ↔ ∃𝑤 𝑤𝑥)
1514anbi2i 445 . . 3 ((𝑥𝐴 ∧ ∃𝑧 𝑧𝑥) ↔ (𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
1615exbii 1537 . 2 (∃𝑥(𝑥𝐴 ∧ ∃𝑧 𝑧𝑥) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
1712, 16sylibr 132 1 (∃𝑦 𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑧 𝑧𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  wss 2984  {csn 3422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428
This theorem is referenced by: (None)
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