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Theorem coss0 34715
Description: Cosets by the empty set are the empty set. (Contributed by Peter Mazsa, 22-Oct-2019.)
Assertion
Ref Expression
coss0 ≀ ∅ = ∅

Proof of Theorem coss0
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfcoss2 34657 . 2 ≀ ∅ = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)}
2 ec0 34617 . . . . . . 7 [𝑥]∅ = ∅
32eleq2i 2868 . . . . . 6 (𝑦 ∈ [𝑥]∅ ↔ 𝑦 ∈ ∅)
42eleq2i 2868 . . . . . 6 (𝑧 ∈ [𝑥]∅ ↔ 𝑧 ∈ ∅)
53, 4anbi12i 621 . . . . 5 ((𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅))
65exbii 1944 . . . 4 (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ ∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅))
7 19.9v 2080 . . . 4 (∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅))
86, 7bitri 267 . . 3 (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅))
98opabbii 4908 . 2 {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)}
10 prnzg 4497 . . . . . 6 (𝑦 ∈ V → {𝑦, 𝑧} ≠ ∅)
1110elv 3387 . . . . 5 {𝑦, 𝑧} ≠ ∅
12 ss0b 4167 . . . . 5 ({𝑦, 𝑧} ⊆ ∅ ↔ {𝑦, 𝑧} = ∅)
1311, 12nemtbir 3064 . . . 4 ¬ {𝑦, 𝑧} ⊆ ∅
14 prssg 4536 . . . . 5 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅))
1514el2v 34483 . . . 4 ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅)
1613, 15mtbir 315 . . 3 ¬ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)
1716opabf 34616 . 2 {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)} = ∅
181, 9, 173eqtri 2823 1 ≀ ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  wne 2969  Vcvv 3383  wss 3767  c0 4113  {cpr 4368  {copab 4903  [cec 7978  ccoss 34461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-xp 5316  df-cnv 5318  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-ec 7982  df-coss 34655
This theorem is referenced by: (None)
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