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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 4796 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
3 | dfopg 4768 | . . . . 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 2833 | . . 3 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ∈ {{𝐴}, {𝐴, 𝐵}}) |
6 | elpri 4549 | . . 3 ⊢ (∅ ∈ {{𝐴}, {𝐴, 𝐵}} → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) |
8 | 2 | simpld 498 | . . . . . 6 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → 𝐴 ∈ V) |
9 | 8 | snn0d 4677 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → {𝐴} ≠ ∅) |
10 | 9 | necomd 2987 | . . . 4 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ≠ {𝐴}) |
11 | prnzg 4680 | . . . . . 6 ⊢ (𝐴 ∈ V → {𝐴, 𝐵} ≠ ∅) | |
12 | 8, 11 | syl 17 | . . . . 5 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → {𝐴, 𝐵} ≠ ∅) |
13 | 12 | necomd 2987 | . . . 4 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ∅ ≠ {𝐴, 𝐵}) |
14 | 10, 13 | jca 515 | . . 3 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → (∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵})) |
15 | neanior 3024 | . . 3 ⊢ ((∅ ≠ {𝐴} ∧ ∅ ≠ {𝐴, 𝐵}) ↔ ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) | |
16 | 14, 15 | sylib 221 | . 2 ⊢ (∅ ∈ 〈𝐴, 𝐵〉 → ¬ (∅ = {𝐴} ∨ ∅ = {𝐴, 𝐵})) |
17 | 7, 16 | pm2.65i 197 | 1 ⊢ ¬ ∅ ∈ 〈𝐴, 𝐵〉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 Vcvv 3398 ∅c0 4223 {csn 4527 {cpr 4529 〈cop 4533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-v 3400 df-dif 3856 df-un 3858 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 |
This theorem is referenced by: opwo0id 5365 0nelelxp 5571 |
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