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 4832 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | dfopg 4804 | . . . . 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 2918 | . . 3 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ∈ {{𝐴}, {𝐴, 𝐵}}) |
6 | elpri 4592 | . . 3 ⊢ (∅ ∈ {{𝐴}, {𝐴, 𝐵}} → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) |
8 | 2 | simpld 497 | . . . . . 6 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → 𝐴 ∈ V) |
9 | snnzg 4713 | . . . . . 6 ⊢ (𝐴 ∈ V → {𝐴} ≠ ∅) | |
10 | 8, 9 | syl 17 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → {𝐴} ≠ ∅) |
11 | 10 | necomd 3074 | . . . 4 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ≠ {𝐴}) |
12 | prnzg 4716 | . . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅) | |
13 | 8, 12 | syl 17 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → {𝐴, 𝐵} ≠ ∅) |
14 | 13 | necomd 3074 | . . . 4 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ≠ {𝐴, 𝐵}) |
15 | 11, 14 | jca 514 | . . 3 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵})) |
16 | neanior 3112 | . . 3 ⊢ ((∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}) ↔ ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) | |
17 | 15, 16 | sylib 220 | . 2 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) |
18 | 7, 17 | pm2.65i 196 | 1 ⊢ ¬ ∅ ∈ 〈𝐴, 𝐵〉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 Vcvv 3497 ∅c0 4294 {csn 4570 {cpr 4572 〈cop 4576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 |
This theorem is referenced by: opwo0id 5390 0nelelxp 5593 |
Copyright terms: Public domain | W3C validator |