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Mirrors > Home > ILE Home > Th. List > mss | GIF version |
Description: An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.) |
Ref | Expression |
---|---|
mss | ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ ∃𝑧 𝑧 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2636 | . . . . 5 ⊢ 𝑦 ∈ V | |
2 | 1 | snss 3588 | . . . 4 ⊢ (𝑦 ∈ 𝐴 ↔ {𝑦} ⊆ 𝐴) |
3 | 1 | snm 3582 | . . . . 5 ⊢ ∃𝑤 𝑤 ∈ {𝑦} |
4 | 1 | snex 4041 | . . . . . 6 ⊢ {𝑦} ∈ V |
5 | sseq1 3062 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → (𝑥 ⊆ 𝐴 ↔ {𝑦} ⊆ 𝐴)) | |
6 | eleq2 2158 | . . . . . . . 8 ⊢ (𝑥 = {𝑦} → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ {𝑦})) | |
7 | 6 | exbidv 1760 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → (∃𝑤 𝑤 ∈ 𝑥 ↔ ∃𝑤 𝑤 ∈ {𝑦})) |
8 | 5, 7 | anbi12d 458 | . . . . . 6 ⊢ (𝑥 = {𝑦} → ((𝑥 ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ 𝑥) ↔ ({𝑦} ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ {𝑦}))) |
9 | 4, 8 | spcev 2727 | . . . . 5 ⊢ (({𝑦} ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ {𝑦}) → ∃𝑥(𝑥 ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ 𝑥)) |
10 | 3, 9 | mpan2 417 | . . . 4 ⊢ ({𝑦} ⊆ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ 𝑥)) |
11 | 2, 10 | sylbi 120 | . . 3 ⊢ (𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ 𝑥)) |
12 | 11 | exlimiv 1541 | . 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ 𝑥)) |
13 | elequ1 1654 | . . . . 5 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝑥 ↔ 𝑤 ∈ 𝑥)) | |
14 | 13 | cbvexv 1850 | . . . 4 ⊢ (∃𝑧 𝑧 ∈ 𝑥 ↔ ∃𝑤 𝑤 ∈ 𝑥) |
15 | 14 | anbi2i 446 | . . 3 ⊢ ((𝑥 ⊆ 𝐴 ∧ ∃𝑧 𝑧 ∈ 𝑥) ↔ (𝑥 ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ 𝑥)) |
16 | 15 | exbii 1548 | . 2 ⊢ (∃𝑥(𝑥 ⊆ 𝐴 ∧ ∃𝑧 𝑧 ∈ 𝑥) ↔ ∃𝑥(𝑥 ⊆ 𝐴 ∧ ∃𝑤 𝑤 ∈ 𝑥)) |
17 | 12, 16 | sylibr 133 | 1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑥(𝑥 ⊆ 𝐴 ∧ ∃𝑧 𝑧 ∈ 𝑥)) |
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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