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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 3367 | . 2 ⊢ (𝐴 ∩ V) ⊆ 𝐴 | |
2 | ssid 3187 | . . 3 ⊢ 𝐴 ⊆ 𝐴 | |
3 | ssv 3189 | . . 3 ⊢ 𝐴 ⊆ V | |
4 | 2, 3 | ssini 3370 | . 2 ⊢ 𝐴 ⊆ (𝐴 ∩ V) |
5 | 1, 4 | eqssi 3183 | 1 ⊢ (𝐴 ∩ V) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1363 Vcvv 2749 ∩ cin 3140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-in 3147 df-ss 3154 |
This theorem is referenced by: rint0 3895 riin0 3970 xpssres 4954 resdmdfsn 4962 imainrect 5086 xpima2m 5088 dmresv 5099 |
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