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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.) Reduce axiom usage and shorten proof. (Revised by Gino Giotto, 3-Oct-2024.) |
Ref | Expression |
---|---|
0nelopab | ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3435 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | vex 3435 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 5389 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ≠ ∅ |
4 | 3 | nesymi 3001 | . . . . 5 ⊢ ¬ ∅ = 〈𝑥, 𝑦〉 |
5 | 4 | intnanr 488 | . . . 4 ⊢ ¬ (∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) |
6 | 5 | nex 1803 | . . 3 ⊢ ¬ ∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) |
7 | 6 | nex 1803 | . 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑) |
8 | elopab 5439 | . 2 ⊢ (∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ ∃𝑥∃𝑦(∅ = 〈𝑥, 𝑦〉 ∧ 𝜑)) | |
9 | 7, 8 | mtbir 323 | 1 ⊢ ¬ ∅ ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∅c0 4258 〈cop 4569 {copab 5137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-v 3433 df-dif 3891 df-un 3893 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-opab 5138 |
This theorem is referenced by: brabv 5479 epelg 5493 satf0n0 33327 bj-0nelmpt 35274 |
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