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Theorem bj-axempty 14916
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 4140. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4141 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 14915 . 2 𝑥𝑦(𝑦𝑥 → ⊥)
2 df-ral 2470 . . 3 (∀𝑦𝑥 ⊥ ↔ ∀𝑦(𝑦𝑥 → ⊥))
32exbii 1615 . 2 (∃𝑥𝑦𝑥 ⊥ ↔ ∃𝑥𝑦(𝑦𝑥 → ⊥))
41, 3mpbir 146 1 𝑥𝑦𝑥
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1361  wfal 1368  wex 1502  wral 2465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-ial 1544  ax-bd0 14836  ax-bdim 14837  ax-bdn 14840  ax-bdeq 14843  ax-bdsep 14907
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-ral 2470
This theorem is referenced by: (None)
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