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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 4188 | . . 3 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
3 | nfopab1 4005 | . . . . . 6 ⊢ Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ 𝜑} | |
4 | 3 | nfel2 2295 | . . . . 5 ⊢ Ⅎ𝑥∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
5 | 4 | nfn 1637 | . . . 4 ⊢ Ⅎ𝑥 ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
6 | nfopab2 4006 | . . . . . . 7 ⊢ Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ 𝜑} | |
7 | 6 | nfel2 2295 | . . . . . 6 ⊢ Ⅎ𝑦∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
8 | 7 | nfn 1637 | . . . . 5 ⊢ Ⅎ𝑦 ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
9 | vex 2692 | . . . . . . . 8 ⊢ 𝑥 ∈ V | |
10 | vex 2692 | . . . . . . . 8 ⊢ 𝑦 ∈ V | |
11 | 9, 10 | opnzi 4165 | . . . . . . 7 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
12 | nesym 2354 | . . . . . . . 8 ⊢ (〈𝑥, 𝑦〉 ≠ ∅ ↔ ¬ ∅ = 〈𝑥, 𝑦〉) | |
13 | pm2.21 607 | . . . . . . . 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 1574 | . . . 4 ⊢ (∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) → ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) |
18 | 5, 17 | exlimi 1574 | . . 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 629 | 1 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ≠ wne 2309 ∅c0 3368 〈cop 3535 {copab 3996 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998 |
This theorem is referenced by: (None) |
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