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Mirrors > Home > ILE Home > Th. List > ssex | GIF version |
Description: The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22. This is one way to express the Axiom of Separation ax-sep 4094 (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994.) |
Ref | Expression |
---|---|
ssex.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
ssex | ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ss 3124 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
2 | ssex.1 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 2 | inex2 4111 | . . 3 ⊢ (𝐴 ∩ 𝐵) ∈ V |
4 | eleq1 2227 | . . 3 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → ((𝐴 ∩ 𝐵) ∈ V ↔ 𝐴 ∈ V)) | |
5 | 3, 4 | mpbii 147 | . 2 ⊢ ((𝐴 ∩ 𝐵) = 𝐴 → 𝐴 ∈ V) |
6 | 1, 5 | sylbi 120 | 1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ∩ cin 3110 ⊆ wss 3111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-in 3117 df-ss 3124 |
This theorem is referenced by: ssexi 4114 ssexg 4115 inteximm 4122 funimaexglem 5265 tfrexlem 6293 elinp 7406 suplocexprlem2b 7646 negfi 11155 ssomct 12315 ssnnctlemct 12316 nninfdc 12325 elcncf 13101 exmid1stab 13714 sbthom 13739 |
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