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Mirrors > Home > MPE Home > Th. List > pssnel | Structured version Visualization version 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 3899 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) | |
2 | n0 3890 | . . 3 ⊢ ((𝐵 ∖ 𝐴) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴)) | |
3 | 1, 2 | sylib 207 | . 2 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴)) |
4 | eldif 3550 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
5 | 4 | exbii 1764 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) |
6 | 3, 5 | sylib 207 | 1 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 ⊊ wpss 3541 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 |
This theorem is referenced by: php 8029 php3 8031 pssnn 8063 inf3lem2 8409 infpssr 9013 ssfin4 9015 genpnnp 9706 ltexprlem1 9737 reclem2pr 9749 mrieqv2d 16122 lbspss 18903 lsmcv 18962 lidlnz 19049 obslbs 19893 nmoid 22356 spansncvi 27895 lsat0cv 33338 osumcllem11N 34270 pexmidlem8N 34281 isomenndlem 39420 |
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