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Theorem bj-axempty 11784
Description: Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a nonempty universe. See axnul 3964. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3965 instead. (New usage is discouraged.)
Assertion
Ref Expression
bj-axempty 𝑥𝑦𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-axempty
StepHypRef Expression
1 bj-axemptylem 11783 . 2 𝑥𝑦(𝑦𝑥 → ⊥)
2 df-ral 2364 . . 3 (∀𝑦𝑥 ⊥ ↔ ∀𝑦(𝑦𝑥 → ⊥))
32exbii 1541 . 2 (∃𝑥𝑦𝑥 ⊥ ↔ ∃𝑥𝑦(𝑦𝑥 → ⊥))
41, 3mpbir 144 1 𝑥𝑦𝑥
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  wfal 1294  wex 1426  wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472  ax-bd0 11704  ax-bdim 11705  ax-bdn 11708  ax-bdeq 11711  ax-bdsep 11775
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-ral 2364
This theorem is referenced by: (None)
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