Theorem relintabex 44421

Description: If the intersection of a class is a relation, then the class is nonempty. (Contributed by RP, 12-Aug-2020.)

Assertion

Ref	Expression
relintabex	⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑)

Proof of Theorem relintabex

Step	Hyp	Ref	Expression
1		intnex 5309	. . . 4 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ 𝜑} = V)
2		nrelv 5780	. . . . 5 ⊢ ¬ Rel V
3		releq 5757	. . . . 5 ⊢ (∩ {𝑥 ∣ 𝜑} = V → (Rel ∩ {𝑥 ∣ 𝜑} ↔ Rel V))
4	2, 3	mtbiri 330	. . . 4 ⊢ (∩ {𝑥 ∣ 𝜑} = V → ¬ Rel ∩ {𝑥 ∣ 𝜑})
5	1, 4	sylbi 220	. . 3 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V → ¬ Rel ∩ {𝑥 ∣ 𝜑})
6	5	con4i 115	. 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V)
7		intexab 5310	. 2 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V)
8	6, 7	sylibr 237	1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  Vcvv 3450  ∩ cint 4907  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-int 4908  df-opab 5168  df-xp 5661  df-rel 5662

This theorem is used by:  relintab  44423