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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 4703 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | dfopg 4675 | . . . . 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 2869 | . . 3 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ∈ {{𝐴}, {𝐴, 𝐵}}) |
6 | elpri 4463 | . . 3 ⊢ (∅ ∈ {{𝐴}, {𝐴, 𝐵}} → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) |
8 | 2 | simpld 487 | . . . . . 6 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → 𝐴 ∈ V) |
9 | snnzg 4584 | . . . . . 6 ⊢ (𝐴 ∈ V → {𝐴} ≠ ∅) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → {𝐴} ≠ ∅) |
11 | 10 | necomd 3023 | . . . 4 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ≠ {𝐴}) |
12 | prnzg 4587 | . . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅) | |
13 | 8, 12 | syl 17 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → {𝐴, 𝐵} ≠ ∅) |
14 | 13 | necomd 3023 | . . . 4 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ≠ {𝐴, 𝐵}) |
15 | 11, 14 | jca 504 | . . 3 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵})) |
16 | neanior 3061 | . . 3 ⊢ ((∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}) ↔ ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) | |
17 | 15, 16 | sylib 210 | . 2 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) |
18 | 7, 17 | pm2.65i 186 | 1 ⊢ ¬ ∅ ∈ 〈𝐴, 𝐵〉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2050 ≠ wne 2968 Vcvv 3416 ∅c0 4179 {csn 4441 {cpr 4443 〈cop 4447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2751 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-v 3418 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 |
This theorem is referenced by: opwo0id 5243 0nelelxp 5442 |
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