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Mirrors > Home > NFE Home > Th. List > pssnel | GIF version |
Description: A proper subclass has a member in one argument that's not in both. (Contributed by NM, 29-Feb-1996.) |
Ref | Expression |
---|---|
pssnel | ⊢ (A ⊊ B → ∃x(x ∈ B ∧ ¬ x ∈ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pssdif 3612 | . . 3 ⊢ (A ⊊ B → (B ∖ A) ≠ ∅) | |
2 | n0 3559 | . . 3 ⊢ ((B ∖ A) ≠ ∅ ↔ ∃x x ∈ (B ∖ A)) | |
3 | 1, 2 | sylib 188 | . 2 ⊢ (A ⊊ B → ∃x x ∈ (B ∖ A)) |
4 | eldif 3221 | . . 3 ⊢ (x ∈ (B ∖ A) ↔ (x ∈ B ∧ ¬ x ∈ A)) | |
5 | 4 | exbii 1582 | . 2 ⊢ (∃x x ∈ (B ∖ A) ↔ ∃x(x ∈ B ∧ ¬ x ∈ A)) |
6 | 3, 5 | sylib 188 | 1 ⊢ (A ⊊ B → ∃x(x ∈ B ∧ ¬ x ∈ A)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 358 ∃wex 1541 ∈ wcel 1710 ≠ wne 2516 ∖ cdif 3206 ⊊ wpss 3258 ∅c0 3550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-pss 3261 df-nul 3551 |
This theorem is referenced by: (None) |
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