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Theorem relintabex 38407
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 4970 . . . 4 {𝑥𝜑} ∈ V ↔ {𝑥𝜑} = V)
2 0nelxp 5300 . . . . . . 7 ¬ ∅ ∈ (V × V)
3 0ex 4942 . . . . . . . 8 ∅ ∈ V
4 eleq1 2827 . . . . . . . . 9 (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
54notbid 307 . . . . . . . 8 (𝑥 = ∅ → (¬ 𝑥 ∈ (V × V) ↔ ¬ ∅ ∈ (V × V)))
63, 5spcev 3440 . . . . . . 7 (¬ ∅ ∈ (V × V) → ∃𝑥 ¬ 𝑥 ∈ (V × V))
72, 6ax-mp 5 . . . . . 6 𝑥 ¬ 𝑥 ∈ (V × V)
8 nss 3804 . . . . . . . 8 (¬ V ⊆ (V × V) ↔ ∃𝑥(𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V × V)))
9 df-rex 3056 . . . . . . . 8 (∃𝑥 ∈ V ¬ 𝑥 ∈ (V × V) ↔ ∃𝑥(𝑥 ∈ V ∧ ¬ 𝑥 ∈ (V × V)))
10 rexv 3360 . . . . . . . 8 (∃𝑥 ∈ V ¬ 𝑥 ∈ (V × V) ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
118, 9, 103bitr2i 288 . . . . . . 7 (¬ V ⊆ (V × V) ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
12 df-rel 5273 . . . . . . 7 (Rel V ↔ V ⊆ (V × V))
1311, 12xchnxbir 322 . . . . . 6 (¬ Rel V ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
147, 13mpbir 221 . . . . 5 ¬ Rel V
15 releq 5358 . . . . 5 ( {𝑥𝜑} = V → (Rel {𝑥𝜑} ↔ Rel V))
1614, 15mtbiri 316 . . . 4 ( {𝑥𝜑} = V → ¬ Rel {𝑥𝜑})
171, 16sylbi 207 . . 3 {𝑥𝜑} ∈ V → ¬ Rel {𝑥𝜑})
1817con4i 113 . 2 (Rel {𝑥𝜑} → {𝑥𝜑} ∈ V)
19 intexab 4971 . 2 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
2018, 19sylibr 224 1 (Rel {𝑥𝜑} → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wrex 3051  Vcvv 3340  wss 3715  c0 4058   cint 4627   × cxp 5264  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-int 4628  df-opab 4865  df-xp 5272  df-rel 5273
This theorem is referenced by:  relintab  38409
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