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Theorem disjALTV0 36417
 Description: The null class is disjoint. (Contributed by Peter Mazsa, 27-Sep-2021.)
Assertion
Ref Expression
disjALTV0 Disj ∅

Proof of Theorem disjALTV0
Dummy variables 𝑥 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 br0 5082 . . . . 5 ¬ 𝑢𝑥
21nex 1803 . . . 4 ¬ ∃𝑢 𝑢𝑥
3 nexmo 2559 . . . 4 (¬ ∃𝑢 𝑢𝑥 → ∃*𝑢 𝑢𝑥)
42, 3ax-mp 5 . . 3 ∃*𝑢 𝑢𝑥
54ax-gen 1798 . 2 𝑥∃*𝑢 𝑢𝑥
6 rel0 5642 . 2 Rel ∅
7 dfdisjALTV4 36382 . 2 ( Disj ∅ ↔ (∀𝑥∃*𝑢 𝑢𝑥 ∧ Rel ∅))
85, 6, 7mpbir2an 711 1 Disj ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3  ∀wal 1537  ∃wex 1782  ∃*wmo 2556  ∅c0 4226   class class class wbr 5033  Rel wrel 5530   Disj wdisjALTV 35920 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-coss 36092  df-cnvrefrel 36198  df-disjALTV 36371 This theorem is referenced by: (None)
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