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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 3235 | . 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴 | |
2 | ssid 3059 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
3 | ssv 3061 | . . 3 ⊢ 𝐴 ⊆ V | |
4 | 2, 3 | ssini 3238 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V) |
5 | 1, 4 | eqssi 3055 | 1 ⊢ (𝐴 ∩ V) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1296 Vcvv 2633 ∩ cin 3012 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-v 2635 df-in 3019 df-ss 3026 |
This theorem is referenced by: rint0 3749 riin0 3823 xpssres 4780 resdmdfsn 4788 imainrect 4910 xpima2m 4912 dmresv 4923 |
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