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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 4331 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
2 | simprl 770 | . . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅) | |
3 | 1, 2 | mto 196 | . . . . 5 ⊢ ¬ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
4 | 3 | nex 1803 | . . . 4 ⊢ ¬ ∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
5 | 4 | nex 1803 | . . 3 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
6 | elxp 5700 | . . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))) | |
7 | 5, 6 | mtbir 323 | . 2 ⊢ ¬ 𝑧 ∈ (∅ × 𝐴) |
8 | 7 | nel0 4351 | 1 ⊢ (∅ × 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∅c0 4323 ⟨cop 4635 × cxp 5675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-opab 5212 df-xp 5683 |
This theorem is referenced by: dmxpid 5930 csbres 5985 res0 5986 xp0 6158 xpnz 6159 xpdisj1 6161 difxp2 6166 xpcan2 6177 xpima 6182 unixp 6282 unixpid 6284 xpcoid 6290 fodomr 9128 xpfiOLD 9318 iundom2g 10535 hashxplem 14393 dmtrclfv 14965 ramcl 16962 0subcat 17788 mat0dimbas0 21968 mavmul0g 22055 txindislem 23137 txhaus 23151 tmdgsum 23599 ust0 23724 ehl0 24934 mbf0 25151 hashxpe 32019 gsumpart 32207 sibf0 33333 lpadlem3 33690 mexval2 34494 poimirlem5 36493 poimirlem10 36498 poimirlem22 36510 poimirlem23 36511 poimirlem26 36514 poimirlem28 36516 0no 42186 0heALT 42534 |
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