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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 BJ, 3-Apr-2019.) |
Ref | Expression |
---|---|
sels | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snidg 4592 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴}) | |
2 | snex 5325 | . . 3 ⊢ {𝐴} ∈ V | |
3 | eleq2 2900 | . . 3 ⊢ (𝑥 = {𝐴} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝐴})) | |
4 | 2, 3 | spcev 3604 | . 2 ⊢ (𝐴 ∈ {𝐴} → ∃𝑥 𝐴 ∈ 𝑥) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1779 ∈ wcel 2113 {csn 4560 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-v 3493 df-dif 3932 df-un 3934 df-nul 4285 df-sn 4561 df-pr 4563 |
This theorem is referenced by: sat1el2xp 32647 |
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