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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 4175 | . . 3 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
3 | nfopab1 3992 | . . . . . 6 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
4 | 3 | nfel2 2292 | . . . . 5 ⊢ Ⅎ𝑥∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
5 | 4 | nfn 1636 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
6 | nfopab2 3993 | . . . . . . 7 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
7 | 6 | nfel2 2292 | . . . . . 6 ⊢ Ⅎ𝑦∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
8 | 7 | nfn 1636 | . . . . 5 ⊢ Ⅎ𝑦 ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
9 | vex 2684 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
10 | vex 2684 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
11 | 9, 10 | opnzi 4152 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
12 | nesym 2351 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ≠ ∅ ↔ ¬ ∅ = 〈𝑥, 𝑦〉) | |
13 | pm2.21 606 | . . . . . . . 8 ⊢ (¬ ∅ = 〈𝑥, 𝑦〉 → (∅ = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | |
14 | 12, 13 | sylbi 120 | . . . . . . 7 ⊢ (〈𝑥, 𝑦〉 ≠ ∅ → (∅ = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) |
15 | 11, 14 | ax-mp 5 | . . . . . 6 ⊢ (∅ = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
16 | 15 | adantr 274 | . . . . 5 ⊢ ((∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
17 | 8, 16 | exlimi 1573 | . . . 4 ⊢ (∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
18 | 5, 17 | exlimi 1573 | . . 3 ⊢ (∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
19 | 2, 18 | sylbi 120 | . 2 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
20 | 1, 19 | pm2.65i 628 | 1 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ≠ wne 2306 ∅c0 3358 〈cop 3525 {copab 3983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-opab 3985 |
This theorem is referenced by: (None) |
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