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Mirrors > Home > ILE Home > Th. List > ssexg | GIF version |
Description: The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.) |
Ref | Expression |
---|---|
ssexg | ⊢ ((A ⊆ B ∧ B ∈ 𝐶) → A ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 2961 | . . . 4 ⊢ (x = B → (A ⊆ x ↔ A ⊆ B)) | |
2 | 1 | imbi1d 220 | . . 3 ⊢ (x = B → ((A ⊆ x → A ∈ V) ↔ (A ⊆ B → A ∈ V))) |
3 | vex 2554 | . . . 4 ⊢ x ∈ V | |
4 | 3 | ssex 3885 | . . 3 ⊢ (A ⊆ x → A ∈ V) |
5 | 2, 4 | vtoclg 2607 | . 2 ⊢ (B ∈ 𝐶 → (A ⊆ B → A ∈ V)) |
6 | 5 | impcom 116 | 1 ⊢ ((A ⊆ B ∧ B ∈ 𝐶) → A ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1242 ∈ wcel 1390 Vcvv 2551 ⊆ wss 2911 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-in 2918 df-ss 2925 |
This theorem is referenced by: ssexd 3888 difexg 3889 rabexg 3891 elssabg 3893 elpw2g 3901 abssexg 3925 snexgOLD 3926 snexg 3927 sess1 4059 sess2 4060 trsuc 4125 unexb 4143 uniexb 4171 xpexg 4395 riinint 4536 dmexg 4539 rnexg 4540 resexg 4593 resiexg 4596 imaexg 4623 exse2 4642 cnvexg 4798 coexg 4805 fabexg 5020 f1oabexg 5081 relrnfvex 5136 fvexg 5137 sefvex 5139 mptfvex 5199 mptexg 5329 ofres 5667 resfunexgALT 5679 cofunexg 5680 fnexALT 5682 f1dmex 5685 oprabexd 5696 mpt2exxg 5775 tposexg 5814 frecabex 5923 erex 6066 ssdomg 6194 fiprc 6228 |
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