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Mirrors > Home > MPE Home > Th. List > 0neqopab | Structured version Visualization version GIF version |
Description: The empty set is never an element in an ordered-pair class abstraction. (Contributed by Alexander van der Vekens, 5-Nov-2017.) |
Ref | Expression |
---|---|
0neqopab | ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopab 5133 | . . 3 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | nfopab1 4871 | . . . . . 6 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 2 | nfel2 2919 | . . . . 5 ⊢ Ⅎ𝑥∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
4 | 3 | nfn 1933 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
5 | nfopab2 4872 | . . . . . . 7 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
6 | 5 | nfel2 2919 | . . . . . 6 ⊢ Ⅎ𝑦∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
7 | 6 | nfn 1933 | . . . . 5 ⊢ Ⅎ𝑦 ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
8 | vex 3343 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
9 | vex 3343 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | opnzi 5091 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
11 | nesym 2988 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ≠ ∅ ↔ ¬ ∅ = 〈𝑥, 𝑦〉) | |
12 | pm2.21 120 | . . . . . . . 8 ⊢ (¬ ∅ = 〈𝑥, 𝑦〉 → (∅ = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
13 | 11, 12 | sylbi 207 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ≠ ∅ → (∅ = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) |
14 | 10, 13 | ax-mp 5 | . . . . . 6 ⊢ (∅ = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
15 | 14 | adantr 472 | . . . . 5 ⊢ ((∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
16 | 7, 15 | exlimi 2233 | . . . 4 ⊢ (∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
17 | 4, 16 | exlimi 2233 | . . 3 ⊢ (∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
18 | 1, 17 | sylbi 207 | . 2 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
19 | id 22 | . 2 ⊢ (¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
20 | 18, 19 | pm2.61i 176 | 1 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1632 ∃wex 1853 ∈ wcel 2139 ≠ wne 2932 ∅c0 4058 〈cop 4327 {copab 4864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-v 3342 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-sn 4322 df-pr 4324 df-op 4328 df-opab 4865 |
This theorem is referenced by: brabv 6865 bj-0nelmpt 33393 |
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