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Mirrors > Home > MPE Home > Th. List > sels | Structured version Visualization version GIF version |
Description: If a class is a set, then it is a member of a set. (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5395. (Revised by BJ, 3-Apr-2019.) Avoid ax-sep 5254, ax-nul 5261, ax-pow 5318. (Revised by BTernaryTau, 15-Jan-2025.) |
Ref | Expression |
---|---|
sels | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2825 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
2 | 1 | exbidv 1924 | . 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝐴 ∈ 𝑥)) |
3 | el 5392 | . 2 ⊢ ∃𝑥 𝑦 ∈ 𝑥 | |
4 | 2, 3 | vtoclg 3523 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∃wex 1781 ∈ wcel 2106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 |
This theorem is referenced by: sat1el2xp 33785 |
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