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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 {𝑥𝜑} → ∃𝑥𝜑)

Proof of Theorem relintabex
StepHypRef Expression
1 intnex 5045 . . . 4 {𝑥𝜑} ∈ V ↔ {𝑥𝜑} = V)
2 0nelxp 5380 . . . . . . 7 ¬ ∅ ∈ (V × V)
3 0ex 5016 . . . . . . . 8 ∅ ∈ V
4 eleq1 2894 . . . . . . . . 9 (𝑥 = ∅ → (𝑥 ∈ (V × V) ↔ ∅ ∈ (V × V)))
54notbid 310 . . . . . . . 8 (𝑥 = ∅ → (¬ 𝑥 ∈ (V × V) ↔ ¬ ∅ ∈ (V × V)))
63, 5spcev 3517 . . . . . . 7 (¬ ∅ ∈ (V × V) → ∃𝑥 ¬ 𝑥 ∈ (V × V))
72, 6ax-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))
118, 9, 103bitr2i 291 . . . . . . 7 (¬ V ⊆ (V × V) ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
12 df-rel 5353 . . . . . . 7 (Rel V ↔ V ⊆ (V × V))
1311, 12xchnxbir 325 . . . . . 6 (¬ Rel V ↔ ∃𝑥 ¬ 𝑥 ∈ (V × V))
147, 13mpbir 223 . . . . 5 ¬ Rel V
15 releq 5440 . . . . 5 ( {𝑥𝜑} = V → (Rel {𝑥𝜑} ↔ Rel V))
1614, 15mtbiri 319 . . . 4 ( {𝑥𝜑} = V → ¬ Rel {𝑥𝜑})
171, 16sylbi 209 . . 3 {𝑥𝜑} ∈ V → ¬ Rel {𝑥𝜑})
1817con4i 114 . 2 (Rel {𝑥𝜑} → {𝑥𝜑} ∈ V)
19 intexab 5046 . 2 (∃𝑥𝜑 {𝑥𝜑} ∈ V)
2018, 19sylibr 226 1 (Rel {𝑥𝜑} → ∃𝑥𝜑)
Colors of variables: wff setvar class
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
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