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Theorem coss0 34552
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 34494 . 2 ≀ ∅ = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)}
2 ec0 34454 . . . . . . 7 [𝑥]∅ = ∅
32eleq2i 2831 . . . . . 6 (𝑦 ∈ [𝑥]∅ ↔ 𝑦 ∈ ∅)
42eleq2i 2831 . . . . . 6 (𝑧 ∈ [𝑥]∅ ↔ 𝑧 ∈ ∅)
53, 4anbi12i 735 . . . . 5 ((𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅))
65exbii 1923 . . . 4 (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ ∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅))
7 19.9v 2062 . . . 4 (∃𝑥(𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅))
86, 7bitri 264 . . 3 (∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅) ↔ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅))
98opabbii 4869 . 2 {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦 ∈ [𝑥]∅ ∧ 𝑧 ∈ [𝑥]∅)} = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)}
10 prnzg 4454 . . . . . 6 (𝑦 ∈ V → {𝑦, 𝑧} ≠ ∅)
1110elv 34309 . . . . 5 {𝑦, 𝑧} ≠ ∅
12 ss0b 4116 . . . . 5 ({𝑦, 𝑧} ⊆ ∅ ↔ {𝑦, 𝑧} = ∅)
1311, 12nemtbir 3027 . . . 4 ¬ {𝑦, 𝑧} ⊆ ∅
14 prssg 4495 . . . . 5 ((𝑦 ∈ V ∧ 𝑧 ∈ V) → ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅))
1514el2v 34310 . . . 4 ((𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅) ↔ {𝑦, 𝑧} ⊆ ∅)
1613, 15mtbir 312 . . 3 ¬ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)
1716opabf 34453 . 2 {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ ∅ ∧ 𝑧 ∈ ∅)} = ∅
181, 9, 173eqtri 2786 1 ≀ ∅ = ∅
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2139  wne 2932  Vcvv 3340  wss 3715  c0 4058  {cpr 4323  {copab 4864  [cec 7909  ccoss 34296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ec 7913  df-coss 34492
This theorem is referenced by: (None)
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