Theorem relintabex 38387

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

Assertion

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

Proof of Theorem relintabex

Step	Hyp	Ref	Expression
1		intnex 4968	. . . 4 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ 𝜑} = V)
2		0nelxp 5298	. . . . . . 7 ⊢ ¬ ∅ ∈ (V × V)
3		0ex 4940	. . . . . . . 8 ⊢ ∅ ∈ V
4		eleq1 2825	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
5	4	notbid 307	. . . . . . . 8 ⊢ (𝑥 = ∅ → (¬ 𝑥 ∈ (V × V) ↔ ¬ ∅ ∈ (V × V)))
6	3, 5	spcev 3438	. . . . . . 7 ⊢ (¬ ∅ ∈ (V × V) → ∃𝑥 ¬ 𝑥 ∈ (V × V))
7	2, 6	ax-mp 5	. . . . . 6 ⊢ ∃𝑥 ¬ 𝑥 ∈ (V × V)
8		nss 3802	. . . . . . . 8 ⊢ (¬ V ⊆ (V × V) ↔ ∃𝑥(𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V × V)))
9		df-rex 3054	. . . . . . . 8 ⊢ (∃𝑥 ∈ V ¬ 𝑥 ∈ (V × V) ↔ ∃𝑥(𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V × V)))
10		rexv 3358	. . . . . . . 8 ⊢ (∃𝑥 ∈ V ¬ 𝑥 ∈ (V × V) ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
11	8, 9, 10	3bitr2i 288	. . . . . . 7 ⊢ (¬ V ⊆ (V × V) ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
12		df-rel 5271	. . . . . . 7 ⊢ (Rel V ↔ V ⊆ (V × V))
13	11, 12	xchnxbir 322	. . . . . 6 ⊢ (¬ Rel V ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
14	7, 13	mpbir 221	. . . . 5 ⊢ ¬ Rel V
15		releq 5356	. . . . 5 ⊢ (∩ {𝑥 ∣ 𝜑} = V → (Rel ∩ {𝑥 ∣ 𝜑} ↔ Rel V))
16	14, 15	mtbiri 316	. . . 4 ⊢ (∩ {𝑥 ∣ 𝜑} = V → ¬ Rel ∩ {𝑥 ∣ 𝜑})
17	1, 16	sylbi 207	. . 3 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V → ¬ Rel ∩ {𝑥 ∣ 𝜑})
18	17	con4i 113	. 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V)
19		intexab 4969	. 2 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V)
20	18, 19	sylibr 224	1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1630  ∃wex 1851   ∈ wcel 2137  {cab 2744  ∃wrex 3049  Vcvv 3338   ⊆ wss 3713  ∅c0 4056  ∩ cint 4625   × cxp 5262  Rel wrel 5269

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pr 5053

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-v 3340  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-sn 4320  df-pr 4322  df-op 4326  df-int 4626  df-opab 4863  df-xp 5270  df-rel 5271

This theorem is referenced by:  relintab  38389