Theorem relintabex 43248

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 5345	. . . 4 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ 𝜑} = V)
2		nrelv 5806	. . . . 5 ⊢ ¬ Rel V
3		releq 5782	. . . . 5 ⊢ (∩ {𝑥 ∣ 𝜑} = V → (Rel ∩ {𝑥 ∣ 𝜑} ↔ Rel V))
4	2, 3	mtbiri 326	. . . 4 ⊢ (∩ {𝑥 ∣ 𝜑} = V → ¬ Rel ∩ {𝑥 ∣ 𝜑})
5	1, 4	sylbi 216	. . 3 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V → ¬ Rel ∩ {𝑥 ∣ 𝜑})
6	5	con4i 114	. 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V)
7		intexab 5346	. 2 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V)
8	6, 7	sylibr 233	1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2703  Vcvv 3462  ∩ cint 4954  Rel wrel 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-int 4955  df-opab 5216  df-xp 5688  df-rel 5689

This theorem is referenced by:  relintab  43250