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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 4280 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) | |
2 | n0 4260 | . . 3 ⊢ ((𝐵 ∖ 𝐴) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴)) | |
3 | 1, 2 | sylib 221 | . 2 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴)) |
4 | eldif 3891 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
5 | 4 | exbii 1849 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) |
6 | 3, 5 | sylib 221 | 1 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 ⊊ wpss 3882 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 |
This theorem is referenced by: php 8685 php3 8687 pssnn 8720 inf3lem2 9076 infpssr 9719 ssfin4 9721 genpnnp 10416 ltexprlem1 10447 reclem2pr 10459 mrieqv2d 16902 lbspss 19847 lsmcv 19906 lidlnz 19994 obslbs 20419 nmoid 23348 spansncvi 29435 fvineqsneq 34829 lsat0cv 36329 osumcllem11N 37262 pexmidlem8N 37273 isomenndlem 43169 |
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