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Mirrors > Home > MPE Home > Th. List > 0nelopab | 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 |
---|---|
0nelopab | ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopab 5304 | . . 3 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
2 | nfopab1 5031 | . . . . . 6 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
3 | 2 | nfel2 2965 | . . . . 5 ⊢ Ⅎ𝑥∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
4 | 3 | nfn 1838 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
5 | nfopab2 5032 | . . . . . . 7 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
6 | 5 | nfel2 2965 | . . . . . 6 ⊢ Ⅎ𝑦∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
7 | 6 | nfn 1838 | . . . . 5 ⊢ Ⅎ𝑦 ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
8 | vex 3440 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
9 | vex 3440 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
10 | 8, 9 | opnzi 5258 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
11 | nesym 3040 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ≠ ∅ ↔ ¬ ∅ = 〈𝑥, 𝑦〉) | |
12 | pm2.21 123 | . . . . . . . 8 ⊢ (¬ ∅ = 〈𝑥, 𝑦〉 → (∅ = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
13 | 11, 12 | sylbi 218 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ≠ ∅ → (∅ = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) |
14 | 10, 13 | ax-mp 5 | . . . . . 6 ⊢ (∅ = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
15 | 14 | adantr 481 | . . . . 5 ⊢ ((∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
16 | 7, 15 | exlimi 2182 | . . . 4 ⊢ (∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
17 | 4, 16 | exlimi 2182 | . . 3 ⊢ (∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
18 | 1, 17 | sylbi 218 | . 2 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
19 | id 22 | . 2 ⊢ (¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
20 | 18, 19 | pm2.61i 183 | 1 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1522 ∃wex 1761 ∈ wcel 2081 ≠ wne 2984 ∅c0 4211 〈cop 4478 {copab 5024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-opab 5025 |
This theorem is referenced by: brabv 5341 epelg 5354 satf0n0 32233 bj-0nelmpt 34006 |
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