Theorem relintabex 38727

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 5045	. . . 4 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ 𝜑} = V)
2		0nelxp 5380	. . . . . . 7 ⊢ ¬ ∅ ∈ (V × V)
3		0ex 5016	. . . . . . . 8 ⊢ ∅ ∈ V
4		eleq1 2894	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
5	4	notbid 310	. . . . . . . 8 ⊢ (𝑥 = ∅ → (¬ 𝑥 ∈ (V × V) ↔ ¬ ∅ ∈ (V × V)))
6	3, 5	spcev 3517	. . . . . . 7 ⊢ (¬ ∅ ∈ (V × V) → ∃𝑥 ¬ 𝑥 ∈ (V × V))
7	2, 6	ax-mp 5	. . . . . 6 ⊢ ∃𝑥 ¬ 𝑥 ∈ (V × V)
8		nss 3888	. . . . . . . 8 ⊢ (¬ V ⊆ (V × V) ↔ ∃𝑥(𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V × V)))
9		df-rex 3123	. . . . . . . 8 ⊢ (∃𝑥 ∈ V ¬ 𝑥 ∈ (V × V) ↔ ∃𝑥(𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V × V)))
10		rexv 3437	. . . . . . . 8 ⊢ (∃𝑥 ∈ V ¬ 𝑥 ∈ (V × V) ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
11	8, 9, 10	3bitr2i 291	. . . . . . 7 ⊢ (¬ V ⊆ (V × V) ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
12		df-rel 5353	. . . . . . 7 ⊢ (Rel V ↔ V ⊆ (V × V))
13	11, 12	xchnxbir 325	. . . . . 6 ⊢ (¬ Rel V ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
14	7, 13	mpbir 223	. . . . 5 ⊢ ¬ Rel V
15		releq 5440	. . . . 5 ⊢ (∩ {𝑥 ∣ 𝜑} = V → (Rel ∩ {𝑥 ∣ 𝜑} ↔ Rel V))
16	14, 15	mtbiri 319	. . . 4 ⊢ (∩ {𝑥 ∣ 𝜑} = V → ¬ Rel ∩ {𝑥 ∣ 𝜑})
17	1, 16	sylbi 209	. . 3 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V → ¬ Rel ∩ {𝑥 ∣ 𝜑})
18	17	con4i 114	. 2 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∩ {𝑥 ∣ 𝜑} ∈ V)
19		intexab 5046	. 2 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V)
20	18, 19	sylibr 226	1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1656  ∃wex 1878   ∈ wcel 2164  {cab 2811  ∃wrex 3118  Vcvv 3414   ⊆ wss 3798  ∅c0 4146  ∩ cint 4699   × cxp 5344  Rel wrel 5351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-int 4700  df-opab 4938  df-xp 5352  df-rel 5353

This theorem is referenced by:  relintab  38729