![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0nelop | Structured version Visualization version GIF version |
Description: A property of ordered pairs. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
0nelop | ⊢ ¬ ∅ ∈ ⟨𝐴, 𝐵⟩ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . 4 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ∈ ⟨𝐴, 𝐵⟩) | |
2 | oprcl 4902 | . . . . 5 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | dfopg 4874 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}}) |
5 | 1, 4 | eleqtrd 2830 | . . 3 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ∈ {{𝐴}, {𝐴, 𝐵}}) |
6 | elpri 4653 | . . 3 ⊢ (∅ ∈ {{𝐴}, {𝐴, 𝐵}} → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) |
8 | 2 | simpld 493 | . . . . . 6 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → 𝐴 ∈ V) |
9 | 8 | snn0d 4782 | . . . . 5 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → {𝐴} ≠ ∅) |
10 | 9 | necomd 2992 | . . . 4 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ≠ {𝐴}) |
11 | prnzg 4785 | . . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅) | |
12 | 8, 11 | syl 17 | . . . . 5 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → {𝐴, 𝐵} ≠ ∅) |
13 | 12 | necomd 2992 | . . . 4 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ∅ ≠ {𝐴, 𝐵}) |
14 | 10, 13 | jca 510 | . . 3 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → (∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵})) |
15 | neanior 3031 | . . 3 ⊢ ((∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}) ↔ ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) | |
16 | 14, 15 | sylib 217 | . 2 ⊢ (∅ ∈ ⟨𝐴, 𝐵⟩ → ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) |
17 | 7, 16 | pm2.65i 193 | 1 ⊢ ¬ ∅ ∈ ⟨𝐴, 𝐵⟩ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ≠ wne 2936 Vcvv 3471 ∅c0 4324 {csn 4630 {cpr 4632 ⟨cop 4636 |
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 2698 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2937 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 |
This theorem is referenced by: opwo0id 5501 0nelelxp 5715 |
Copyright terms: Public domain | W3C validator |