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Theorem mss 4077
 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 2636 . . . . 5 𝑦 ∈ V
21snss 3588 . . . 4 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
31snm 3582 . . . . 5 𝑤 𝑤 ∈ {𝑦}
41snex 4041 . . . . . 6 {𝑦} ∈ V
5 sseq1 3062 . . . . . . 7 (𝑥 = {𝑦} → (𝑥𝐴 ↔ {𝑦} ⊆ 𝐴))
6 eleq2 2158 . . . . . . . 8 (𝑥 = {𝑦} → (𝑤𝑥𝑤 ∈ {𝑦}))
76exbidv 1760 . . . . . . 7 (𝑥 = {𝑦} → (∃𝑤 𝑤𝑥 ↔ ∃𝑤 𝑤 ∈ {𝑦}))
85, 7anbi12d 458 . . . . . 6 (𝑥 = {𝑦} → ((𝑥𝐴 ∧ ∃𝑤 𝑤𝑥) ↔ ({𝑦} ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ {𝑦})))
94, 8spcev 2727 . . . . 5 (({𝑦} ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ {𝑦}) → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
103, 9mpan2 417 . . . 4 ({𝑦} ⊆ 𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
112, 10sylbi 120 . . 3 (𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
1211exlimiv 1541 . 2 (∃𝑦 𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
13 elequ1 1654 . . . . 5 (𝑧 = 𝑤 → (𝑧𝑥𝑤𝑥))
1413cbvexv 1850 . . . 4 (∃𝑧 𝑧𝑥 ↔ ∃𝑤 𝑤𝑥)
1514anbi2i 446 . . 3 ((𝑥𝐴 ∧ ∃𝑧 𝑧𝑥) ↔ (𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
1615exbii 1548 . 2 (∃𝑥(𝑥𝐴 ∧ ∃𝑧 𝑧𝑥) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
1712, 16sylibr 133 1 (∃𝑦 𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑧 𝑧𝑥))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1296  ∃wex 1433   ∈ wcel 1445   ⊆ wss 3013  {csn 3466 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472 This theorem is referenced by: (None)
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