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Theorem relintabex 36906
 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 {𝑥𝜑} → ∃𝑥𝜑)

Proof of Theorem relintabex
StepHypRef Expression
1 intnex 4748 . . . 4 {𝑥𝜑} ∈ V ↔ {𝑥𝜑} = V)
2 0nelxp 5067 . . . . . . 7 ¬ ∅ ∈ (V × V)
3 0ex 4718 . . . . . . . 8 ∅ ∈ V
4 eleq1 2676 . . . . . . . . 9 (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
54notbid 307 . . . . . . . 8 (𝑥 = ∅ → (¬ 𝑥 ∈ (V × V) ↔ ¬ ∅ ∈ (V × V)))
63, 5spcev 3273 . . . . . . 7 (¬ ∅ ∈ (V × V) → ∃𝑥 ¬ 𝑥 ∈ (V × V))
72, 6ax-mp 5 . . . . . 6 𝑥 ¬ 𝑥 ∈ (V × V)
8 nss 3626 . . . . . . . 8 (¬ V ⊆ (V × V) ↔ ∃𝑥(𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V × V)))
9 df-rex 2902 . . . . . . . 8 (∃𝑥 ∈ V ¬ 𝑥 ∈ (V × V) ↔ ∃𝑥(𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V × V)))
10 rexv 3193 . . . . . . . 8 (∃𝑥 ∈ V ¬ 𝑥 ∈ (V × V) ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
118, 9, 103bitr2i 287 . . . . . . 7 (¬ V ⊆ (V × V) ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
12 df-rel 5045 . . . . . . 7 (Rel V ↔ V ⊆ (V × V))
1311, 12xchnxbir 322 . . . . . 6 (¬ Rel V ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
147, 13mpbir 220 . . . . 5 ¬ Rel V
15 releq 5124 . . . . 5 ( {𝑥𝜑} = V → (Rel {𝑥𝜑} ↔ Rel V))
1614, 15mtbiri 316 . . . 4 ( {𝑥𝜑} = V → ¬ Rel {𝑥𝜑})
171, 16sylbi 206 . . 3 {𝑥𝜑} ∈ V → ¬ Rel {𝑥𝜑})
1817con4i 112 . 2 (Rel {𝑥𝜑} → {𝑥𝜑} ∈ V)
19 intexab 4749 . 2 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
2018, 19sylibr 223 1 (Rel {𝑥𝜑} → ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ∩ cint 4410   × cxp 5036  Rel wrel 5043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-int 4411  df-opab 4644  df-xp 5044  df-rel 5045 This theorem is referenced by:  relintab  36908
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