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Theorem cusgrasizeindslem1 26002
 Description: Lemma 1 for cusgrasizeinds 26004. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
Hypothesis
Ref Expression
cusgrares.f 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
Assertion
Ref Expression
cusgrasizeindslem1 (dom 𝐹 ∩ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ∅
Distinct variable groups:   𝑥,𝐸   𝑥,𝑁
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem cusgrasizeindslem1
StepHypRef Expression
1 cusgrares.f . . . . 5 𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
21dmeqi 5247 . . . 4 dom 𝐹 = dom (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
3 incom 3767 . . . . 5 ({𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∩ dom 𝐸) = (dom 𝐸 ∩ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
4 dmres 5339 . . . . 5 dom (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) = ({𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∩ dom 𝐸)
5 ssid 3587 . . . . . 6 dom 𝐸 ⊆ dom 𝐸
6 dfrab3ss 3864 . . . . . 6 (dom 𝐸 ⊆ dom 𝐸 → {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} = (dom 𝐸 ∩ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}))
75, 6ax-mp 5 . . . . 5 {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} = (dom 𝐸 ∩ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})
83, 4, 73eqtr4i 2642 . . . 4 dom (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}) = {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}
92, 8eqtri 2632 . . 3 dom 𝐹 = {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)}
109ineq1i 3772 . 2 (dom 𝐹 ∩ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ({𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∩ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)})
11 inrab 3858 . . 3 ({𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∩ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = {𝑥 ∈ dom 𝐸 ∣ (𝑁 ∉ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑥))}
12 exmid 430 . . . . . . 7 (𝑁 ∈ (𝐸𝑥) ∨ ¬ 𝑁 ∈ (𝐸𝑥))
13 nnel 2892 . . . . . . . 8 𝑁 ∉ (𝐸𝑥) ↔ 𝑁 ∈ (𝐸𝑥))
1413orbi1i 541 . . . . . . 7 ((¬ 𝑁 ∉ (𝐸𝑥) ∨ ¬ 𝑁 ∈ (𝐸𝑥)) ↔ (𝑁 ∈ (𝐸𝑥) ∨ ¬ 𝑁 ∈ (𝐸𝑥)))
1512, 14mpbir 220 . . . . . 6 𝑁 ∉ (𝐸𝑥) ∨ ¬ 𝑁 ∈ (𝐸𝑥))
16 ianor 508 . . . . . 6 (¬ (𝑁 ∉ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑥)) ↔ (¬ 𝑁 ∉ (𝐸𝑥) ∨ ¬ 𝑁 ∈ (𝐸𝑥)))
1715, 16mpbir 220 . . . . 5 ¬ (𝑁 ∉ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑥))
1817rgenw 2908 . . . 4 𝑥 ∈ dom 𝐸 ¬ (𝑁 ∉ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑥))
19 rabeq0 3911 . . . 4 ({𝑥 ∈ dom 𝐸 ∣ (𝑁 ∉ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑥))} = ∅ ↔ ∀𝑥 ∈ dom 𝐸 ¬ (𝑁 ∉ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑥)))
2018, 19mpbir 220 . . 3 {𝑥 ∈ dom 𝐸 ∣ (𝑁 ∉ (𝐸𝑥) ∧ 𝑁 ∈ (𝐸𝑥))} = ∅
2111, 20eqtri 2632 . 2 ({𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)} ∩ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ∅
2210, 21eqtri 2632 1 (dom 𝐹 ∩ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ∀wral 2896  {crab 2900   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  dom cdm 5038   ↾ cres 5040  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-res 5050 This theorem is referenced by:  cusgrasizeinds  26004
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