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Mirrors > Home > ILE Home > Th. List > 0neqopab | 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 | id 19 | . 2 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
2 | elopab 4085 | . . 3 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
3 | nfopab1 3907 | . . . . . 6 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
4 | 3 | nfel2 2241 | . . . . 5 ⊢ Ⅎ𝑥∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
5 | 4 | nfn 1593 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
6 | nfopab2 3908 | . . . . . . 7 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
7 | 6 | nfel2 2241 | . . . . . 6 ⊢ Ⅎ𝑦∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
8 | 7 | nfn 1593 | . . . . 5 ⊢ Ⅎ𝑦 ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
9 | vex 2622 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
10 | vex 2622 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
11 | 9, 10 | opnzi 4062 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
12 | nesym 2300 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ≠ ∅ ↔ ¬ ∅ = 〈𝑥, 𝑦〉) | |
13 | pm2.21 582 | . . . . . . . 8 ⊢ (¬ ∅ = 〈𝑥, 𝑦〉 → (∅ = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
14 | 12, 13 | sylbi 119 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ≠ ∅ → (∅ = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) |
15 | 11, 14 | ax-mp 7 | . . . . . 6 ⊢ (∅ = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
16 | 15 | adantr 270 | . . . . 5 ⊢ ((∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
17 | 8, 16 | exlimi 1530 | . . . 4 ⊢ (∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
18 | 5, 17 | exlimi 1530 | . . 3 ⊢ (∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
19 | 2, 18 | sylbi 119 | . 2 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
20 | 1, 19 | pm2.65i 603 | 1 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 = wceq 1289 ∃wex 1426 ∈ wcel 1438 ≠ wne 2255 ∅c0 3286 〈cop 3449 {copab 3898 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-v 2621 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-opab 3900 |
This theorem is referenced by: (None) |
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