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Mirrors > Home > ILE Home > Th. List > inv1 | GIF version |
Description: The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.) |
Ref | Expression |
---|---|
inv1 | ⊢ (A ∩ V) = A |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inss1 3151 | . 2 ⊢ (A ∩ V) ⊆ A | |
2 | ssid 2958 | . . 3 ⊢ A ⊆ A | |
3 | ssv 2959 | . . 3 ⊢ A ⊆ V | |
4 | 2, 3 | ssini 3154 | . 2 ⊢ A ⊆ (A ∩ V) |
5 | 1, 4 | eqssi 2955 | 1 ⊢ (A ∩ V) = A |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 Vcvv 2551 ∩ cin 2910 |
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 |
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: rint0 3645 riin0 3719 xpssres 4588 imainrect 4709 xpima2m 4711 dmresv 4722 |
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