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Theorem disjALTV0 37216
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 5154 . . . . 5 ¬ 𝑢𝑥
21nex 1802 . . . 4 ¬ ∃𝑢 𝑢𝑥
3 nexmo 2539 . . . 4 (¬ ∃𝑢 𝑢𝑥 → ∃*𝑢 𝑢𝑥)
42, 3ax-mp 5 . . 3 ∃*𝑢 𝑢𝑥
54ax-gen 1797 . 2 𝑥∃*𝑢 𝑢𝑥
6 rel0 5755 . 2 Rel ∅
7 dfdisjALTV4 37178 . 2 ( Disj ∅ ↔ (∀𝑥∃*𝑢 𝑢𝑥 ∧ Rel ∅))
85, 6, 7mpbir2an 709 1 Disj ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wal 1539  wex 1781  ∃*wmo 2536  c0 4282   class class class wbr 5105  Rel wrel 5638   Disj wdisjALTV 36668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-coss 36873  df-cnvrefrel 36989  df-disjALTV 37167
This theorem is referenced by:  eqvrel0  37248  det0  37249  eqvrelcoss0  37250  pet02  37276
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