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 4427 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | dfopg 4400 | . . . . 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 2703 | . . 3 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ∈ {{𝐴}, {𝐴, 𝐵}}) |
6 | elpri 4197 | . . 3 ⊢ (∅ ∈ {{𝐴}, {𝐴, 𝐵}} → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) |
8 | 2 | simpld 475 | . . . . . 6 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → 𝐴 ∈ V) |
9 | snnzg 4308 | . . . . . 6 ⊢ (𝐴 ∈ V → {𝐴} ≠ ∅) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → {𝐴} ≠ ∅) |
11 | 10 | necomd 2849 | . . . 4 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ≠ {𝐴}) |
12 | prnzg 4311 | . . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅) | |
13 | 8, 12 | syl 17 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → {𝐴, 𝐵} ≠ ∅) |
14 | 13 | necomd 2849 | . . . 4 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ≠ {𝐴, 𝐵}) |
15 | 11, 14 | jca 554 | . . 3 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵})) |
16 | neanior 2886 | . . 3 ⊢ ((∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}) ↔ ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) | |
17 | 15, 16 | sylib 208 | . 2 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) |
18 | 7, 17 | pm2.65i 185 | 1 ⊢ ¬ ∅ ∈ 〈𝐴, 𝐵〉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∅c0 3915 {csn 4177 {cpr 4179 〈cop 4183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 |
This theorem is referenced by: opwo0id 4961 0nelelxp 5145 |
Copyright terms: Public domain | W3C validator |