![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0xp | Structured version Visualization version GIF version |
Description: The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
0xp | ⊢ (∅ × 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4247 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | simprl 770 | . . . . . 6 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅) | |
3 | 1, 2 | mto 200 | . . . . 5 ⊢ ¬ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
4 | 3 | nex 1802 | . . . 4 ⊢ ¬ ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
5 | 4 | nex 1802 | . . 3 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
6 | elxp 5542 | . . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))) | |
7 | 5, 6 | mtbir 326 | . 2 ⊢ ¬ 𝑧 ∈ (∅ × 𝐴) |
8 | 7 | nel0 4264 | 1 ⊢ (∅ × 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∅c0 4243 〈cop 4531 × cxp 5517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 |
This theorem is referenced by: dmxpid 5764 csbres 5821 res0 5822 xp0 5982 xpnz 5983 xpdisj1 5985 difxp2 5990 xpcan2 6001 xpima 6006 unixp 6101 unixpid 6103 xpcoid 6109 fodomr 8652 xpfi 8773 iundom2g 9951 hashxplem 13790 dmtrclfv 14369 ramcl 16355 0subcat 17100 mat0dimbas0 21071 mavmul0g 21158 txindislem 22238 txhaus 22252 tmdgsum 22700 ust0 22825 ehl0 24021 mbf0 24238 hashxpe 30555 gsumpart 30740 sibf0 31702 lpadlem3 32059 mexval2 32863 poimirlem5 35062 poimirlem10 35067 poimirlem22 35079 poimirlem23 35080 poimirlem26 35083 poimirlem28 35085 0heALT 40484 |
Copyright terms: Public domain | W3C validator |