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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 3450 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
2 | vex 3450 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
3 | 1, 2 | opnzi 5432 | . . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ≠ ∅ |
4 | 3 | nesymi 3002 | . . . . 5 ⊢ ¬ ∅ = ⟨𝑥, 𝑦⟩ |
5 | 4 | intnanr 489 | . . . 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 5485 | . 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 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∅c0 4283 ⟨cop 4593 {copab 5168 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-opab 5169 |
This theorem is referenced by: brabv 5527 epelg 5539 satf0n0 33975 bj-0nelmpt 35590 |
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