Theorem relintabex 37887

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 4821	. . . 4 ⊢ (¬ ∩ {𝑥 ∣ 𝜑} ∈ V ↔ ∩ {𝑥 ∣ 𝜑} = V)
2		0nelxp 5143	. . . . . . 7 ⊢ ¬ ∅ ∈ (V × V)
3		0ex 4790	. . . . . . . 8 ⊢ ∅ ∈ V
4		eleq1 2689	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
5	4	notbid 308	. . . . . . . 8 ⊢ (𝑥 = ∅ → (¬ 𝑥 ∈ (V × V) ↔ ¬ ∅ ∈ (V × V)))
6	3, 5	spcev 3300	. . . . . . 7 ⊢ (¬ ∅ ∈ (V × V) → ∃𝑥 ¬ 𝑥 ∈ (V × V))
7	2, 6	ax-mp 5	. . . . . 6 ⊢ ∃𝑥 ¬ 𝑥 ∈ (V × V)
8		nss 3663	. . . . . . . 8 ⊢ (¬ V ⊆ (V × V) ↔ ∃𝑥(𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V × V)))
9		df-rex 2918	. . . . . . . 8 ⊢ (∃𝑥 ∈ V ¬ 𝑥 ∈ (V × V) ↔ ∃𝑥(𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V × V)))
10		rexv 3220	. . . . . . . 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 5121	. . . . . . 7 ⊢ (Rel V ↔ V ⊆ (V × V))
13	11, 12	xchnxbir 323	. . . . . 6 ⊢ (¬ Rel V ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
14	7, 13	mpbir 221	. . . . 5 ⊢ ¬ Rel V
15		releq 5201	. . . . 5 ⊢ (∩ {𝑥 ∣ 𝜑} = V → (Rel ∩ {𝑥 ∣ 𝜑} ↔ Rel V))
16	14, 15	mtbiri 317	. . . 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 4822	. 2 ⊢ (∃𝑥𝜑 ↔ ∩ {𝑥 ∣ 𝜑} ∈ V)
20	18, 19	sylibr 224	1 ⊢ (Rel ∩ {𝑥 ∣ 𝜑} → ∃𝑥𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ∩ cint 4475   × cxp 5112  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-int 4476  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  relintab  37889