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Theorem mss 3962
 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 2560 . . . . 5 𝑦 ∈ V
21snss 3494 . . . 4 (𝑦𝐴 ↔ {𝑦} ⊆ 𝐴)
31snm 3488 . . . . 5 𝑤 𝑤 ∈ {𝑦}
4 snexgOLD 3935 . . . . . . 7 (𝑦 ∈ V → {𝑦} ∈ V)
51, 4ax-mp 7 . . . . . 6 {𝑦} ∈ V
6 sseq1 2966 . . . . . . 7 (𝑥 = {𝑦} → (𝑥𝐴 ↔ {𝑦} ⊆ 𝐴))
7 eleq2 2101 . . . . . . . 8 (𝑥 = {𝑦} → (𝑤𝑥𝑤 ∈ {𝑦}))
87exbidv 1706 . . . . . . 7 (𝑥 = {𝑦} → (∃𝑤 𝑤𝑥 ↔ ∃𝑤 𝑤 ∈ {𝑦}))
96, 8anbi12d 442 . . . . . 6 (𝑥 = {𝑦} → ((𝑥𝐴 ∧ ∃𝑤 𝑤𝑥) ↔ ({𝑦} ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ {𝑦})))
105, 9spcev 2647 . . . . 5 (({𝑦} ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ {𝑦}) → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
113, 10mpan2 401 . . . 4 ({𝑦} ⊆ 𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
122, 11sylbi 114 . . 3 (𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
1312exlimiv 1489 . 2 (∃𝑦 𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
14 elequ1 1600 . . . . 5 (𝑧 = 𝑤 → (𝑧𝑥𝑤𝑥))
1514cbvexv 1795 . . . 4 (∃𝑧 𝑧𝑥 ↔ ∃𝑤 𝑤𝑥)
1615anbi2i 430 . . 3 ((𝑥𝐴 ∧ ∃𝑧 𝑧𝑥) ↔ (𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
1716exbii 1496 . 2 (∃𝑥(𝑥𝐴 ∧ ∃𝑧 𝑧𝑥) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑤 𝑤𝑥))
1813, 17sylibr 137 1 (∃𝑦 𝑦𝐴 → ∃𝑥(𝑥𝐴 ∧ ∃𝑧 𝑧𝑥))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  {csn 3375 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-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381 This theorem is referenced by: (None)
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