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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 4904 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | dfopg 4876 | . . . . 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 2841 | . . 3 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ∈ {{𝐴}, {𝐴, 𝐵}}) |
6 | elpri 4654 | . . 3 ⊢ (∅ ∈ {{𝐴}, {𝐴, 𝐵}} → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) |
8 | 2 | simpld 494 | . . . . . 6 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → 𝐴 ∈ V) |
9 | 8 | snn0d 4780 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → {𝐴} ≠ ∅) |
10 | 9 | necomd 2994 | . . . 4 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ≠ {𝐴}) |
11 | prnzg 4783 | . . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅) | |
12 | 8, 11 | syl 17 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → {𝐴, 𝐵} ≠ ∅) |
13 | 12 | necomd 2994 | . . . 4 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ≠ {𝐴, 𝐵}) |
14 | 10, 13 | jca 511 | . . 3 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵})) |
15 | neanior 3033 | . . 3 ⊢ ((∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}) ↔ ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) | |
16 | 14, 15 | sylib 218 | . 2 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) |
17 | 7, 16 | pm2.65i 194 | 1 ⊢ ¬ ∅ ∈ 〈𝐴, 𝐵〉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∅c0 4339 {csn 4631 {cpr 4633 〈cop 4637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 |
This theorem is referenced by: opwo0id 5507 0nelelxp 5724 |
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