Theorem ntrneircomplex 44064

Description: The relative complement of the class 𝑆 exists as a subset of the base set. (Contributed by RP, 26-Jun-2021.)

Hypotheses

Ref	Expression
ntrnei.o	⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
ntrnei.f	⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r	⊢ (𝜑 → 𝐼𝐹𝑁)

Assertion

Ref	Expression
ntrneircomplex	⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Distinct variable groups:   𝑖,𝑗,𝑘   𝑖,𝑙,𝑗   𝑖,𝑚,𝑗

Allowed substitution hints:   𝜑(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐵(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑆(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem ntrneircomplex

Step	Hyp	Ref	Expression
1		ntrnei.o	. . 3 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
2		ntrnei.f	. . 3 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
3		ntrnei.r	. . 3 ⊢ (𝜑 → 𝐼𝐹𝑁)
4	1, 2, 3	ntrneibex 44063	. 2 ⊢ (𝜑 → 𝐵 ∈ V)
5		difssd 4147	. 2 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ⊆ 𝐵)
6	4, 5	sselpwd 5334	1 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478   ∖ cdif 3960  𝒫 cpw 4605   class class class wbr 5148   ↦ cmpt 5231  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433   ↑_m cmap 8865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436

This theorem is referenced by: (None)