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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 3472 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | vex 3472 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 5467 | . . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅ |
4 | 3 | nesymi 2992 | . . . . 5 ⊢ ¬ ∅ = ⟨𝑥, 𝑦⟩ |
5 | 4 | intnanr 487 | . . . 4 ⊢ ¬ (∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
6 | 5 | nex 1794 | . . 3 ⊢ ¬ ∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
7 | 6 | nex 1794 | . 2 ⊢ ¬ ∃𝑥∃𝑦(∅ = ⟨𝑥, 𝑦⟩ ∧ 𝜑) |
8 | elopab 5520 | . 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 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∅c0 4317 ⟨cop 4629 {copab 5203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-opab 5204 |
This theorem is referenced by: brabv 5562 epelg 5574 satf0n0 34897 bj-0nelmpt 36504 |
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