Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3945 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → (𝐵 ∖ 𝐴) ≠ ∅) | |
2 | n0 3931 | . . 3 ⊢ ((𝐵 ∖ 𝐴) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴)) | |
3 | 1, 2 | sylib 208 | . 2 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴)) |
4 | eldif 3584 | . . 3 ⊢ (𝑥 ∈ (𝐵 ∖ 𝐴) ↔ (𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) | |
5 | 4 | exbii 1774 | . 2 ⊢ (∃𝑥 𝑥 ∈ (𝐵 ∖ 𝐴) ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) |
6 | 3, 5 | sylib 208 | 1 ⊢ (𝐴 ⊊ 𝐵 → ∃𝑥(𝑥 ∈ 𝐵 ∧ ¬ 𝑥 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 ⊊ wpss 3575 ∅c0 3915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 |
This theorem is referenced by: php 8144 php3 8146 pssnn 8178 inf3lem2 8526 infpssr 9130 ssfin4 9132 genpnnp 9827 ltexprlem1 9858 reclem2pr 9870 mrieqv2d 16299 lbspss 19082 lsmcv 19141 lidlnz 19228 obslbs 20074 nmoid 22546 spansncvi 28511 lsat0cv 34320 osumcllem11N 35252 pexmidlem8N 35263 isomenndlem 40744 |
Copyright terms: Public domain | W3C validator |