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Mirrors > Home > ILE Home > Th. List > nssr | GIF version |
Description: Negation of subclass relationship. One direction of Exercise 13 of [TakeutiZaring] p. 18. (Contributed by Jim Kingdon, 15-Jul-2018.) |
Ref | Expression |
---|---|
nssr | ⊢ (∃x(x ∈ A ∧ ¬ x ∈ B) → ¬ A ⊆ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exanaliim 1535 | . 2 ⊢ (∃x(x ∈ A ∧ ¬ x ∈ B) → ¬ ∀x(x ∈ A → x ∈ B)) | |
2 | dfss2 2928 | . 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B)) | |
3 | 1, 2 | sylnibr 601 | 1 ⊢ (∃x(x ∈ A ∧ ¬ x ∈ B) → ¬ A ⊆ B) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∀wal 1240 ∃wex 1378 ∈ wcel 1390 ⊆ 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-in1 544 ax-in2 545 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 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-fal 1248 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-in 2918 df-ss 2925 |
This theorem is referenced by: (None) |
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