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Theorem List for Intuitionistic Logic Explorer - 10901-11000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembd0r 10901 A formula equivalent to a bounded one is bounded. Stated with a commuted (compared with bd0 10900) biconditional in the hypothesis, to work better with definitions (
ps is the definiendum that one wants to prove bounded). (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  ph   &    |-  ( ps  <->  ph )   =>    |- BOUNDED  ps
 
Theorembdbi 10902 A biconditional between two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  ph   &    |- BOUNDED  ps   =>    |- BOUNDED  ( ph 
 <->  ps )
 
Theorembdstab 10903 Stability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  ph   =>    |- BOUNDED STAB  ph
 
Theorembddc 10904 Decidability of a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  ph   =>    |- BOUNDED DECID  ph
 
Theorembd3or 10905 A disjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  ph   &    |- BOUNDED  ps   &    |- BOUNDED  ch   =>    |- BOUNDED  ( ph  \/  ps  \/  ch )
 
Theorembd3an 10906 A conjunction of three bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  ph   &    |- BOUNDED  ps   &    |- BOUNDED  ch   =>    |- BOUNDED  ( ph  /\  ps  /\ 
 ch )
 
Theorembdth 10907 A truth (a (closed) theorem) is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
 |-  ph   =>    |- BOUNDED  ph
 
Theorembdtru 10908 The truth value T. is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED T.
 
Theorembdfal 10909 The truth value F. is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED F.
 
Theorembdnth 10910 A falsity is a bounded formula. (Contributed by BJ, 6-Oct-2019.)
 |-  -.  ph   =>    |- BOUNDED  ph
 
TheorembdnthALT 10911 Alternate proof of bdnth 10910 not using bdfal 10909. Then, bdfal 10909 can be proved from this theorem, using fal 1292. The total number of proof steps would be 17 (for bdnthALT 10911) + 3 = 20, which is more than 8 (for bdfal 10909) + 9 (for bdnth 10910) = 17. (Contributed by BJ, 6-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  -.  ph   =>    |- BOUNDED  ph
 
Theorembdxor 10912 The exclusive disjunction of two bounded formulas is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  ph   &    |- BOUNDED  ps   =>    |- BOUNDED  ( ph  \/_  ps )
 
Theorembj-bdcel 10913* Boundedness of a membership formula. (Contributed by BJ, 8-Dec-2019.)
 |- BOUNDED  y  =  A   =>    |- BOUNDED  A  e.  x
 
Theorembdab 10914 Membership in a class defined by class abstraction using a bounded formula, is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  ph   =>    |- BOUNDED  x  e.  { y  |  ph }
 
Theorembdcdeq 10915 Conditional equality of a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  ph   =>    |- BOUNDED CondEq ( x  =  y  ->  ph )
 
6.2.5.2  Bounded classes

In line with our definitions of classes as extensions of predicates, it is useful to define a predicate for bounded classes, which is done in df-bdc 10917. Note that this notion is only a technical device which can be used to shorten proofs of (semantic) boundedness of formulas.

As will be clear by the end of this subsection (see for instance bdop 10951), one can prove the boundedness of any concrete term using only setvars and bounded formulas, for instance,  |- BOUNDED  ph =>  |- BOUNDED 
<. { x  |  ph } ,  ( {
y ,  suc  z }  X.  <. t ,  (/) >.
) >.. The proofs are long since one has to prove boundedness at each step of the construction, without being able to prove general theorems like  |- BOUNDED  A =>  |- BOUNDED  { A }.

 
Syntaxwbdc 10916 Syntax for the predicate BOUNDED.
 wff BOUNDED  A
 
Definitiondf-bdc 10917* Define a bounded class as one such that membership in this class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
 |-  (BOUNDED  A  <->  A. xBOUNDED  x  e.  A )
 
Theorembdceq 10918 Equality property for the predicate BOUNDED. (Contributed by BJ, 3-Oct-2019.)
 |-  A  =  B   =>    |-  (BOUNDED  A 
 <-> BOUNDED  B )
 
Theorembdceqi 10919 A class equal to a bounded one is bounded. Note the use of ax-ext 2065. See also bdceqir 10920. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  A   &    |-  A  =  B   =>    |- BOUNDED  B
 
Theorembdceqir 10920 A class equal to a bounded one is bounded. Stated with a commuted (compared with bdceqi 10919) equality in the hypothesis, to work better with definitions ( B is the definiendum that one wants to prove bounded; see comment of bd0r 10901). (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  A   &    |-  B  =  A   =>    |- BOUNDED  B
 
Theorembdel 10921* The belonging of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
 |-  (BOUNDED  A  -> BOUNDED  x  e.  A )
 
Theorembdeli 10922* Inference associated with bdel 10921. Its converse is bdelir 10923. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  A   =>    |- BOUNDED  x  e.  A
 
Theorembdelir 10923* Inference associated with df-bdc 10917. Its converse is bdeli 10922. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  x  e.  A   =>    |- BOUNDED  A
 
Theorembdcv 10924 A setvar is a bounded class. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  x
 
Theorembdcab 10925 A class defined by class abstraction using a bounded formula is bounded. (Contributed by BJ, 6-Oct-2019.)
 |- BOUNDED  ph   =>    |- BOUNDED  { x  |  ph }
 
Theorembdph 10926 A formula which defines (by class abstraction) a bounded class is bounded. (Contributed by BJ, 6-Oct-2019.)
 |- BOUNDED  { x  |  ph }   =>    |- BOUNDED  ph
 
Theorembds 10927* Boundedness of a formula resulting from implicit substitution in a bounded formula. Note that the proof does not use ax-bdsb 10898; therefore, using implicit instead of explicit substitution when boundedness is important, one might avoid using ax-bdsb 10898. (Contributed by BJ, 19-Nov-2019.)
 |- BOUNDED  ph   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |- BOUNDED  ps
 
Theorembdcrab 10928* A class defined by restricted abstraction from a bounded class and a bounded formula is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  A   &    |- BOUNDED  ph   =>    |- BOUNDED  { x  e.  A  |  ph }
 
Theorembdne 10929 Inequality of two setvars is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  x  =/=  y
 
Theorembdnel 10930* Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  A   =>    |- BOUNDED  x  e/  A
 
Theorembdreu 10931* Boundedness of existential uniqueness.

Remark regarding restricted quantifiers: the formula  A. x  e.  A ph need not be bounded even if 
A and  ph are. Indeed,  _V is bounded by bdcvv 10933, and  |-  ( A. x  e. 
_V ph  <->  A. x ph ) (in minimal propositional calculus), so by bd0 10900, if  A. x  e. 
_V ph were bounded when  ph is bounded, then  A. x ph would be bounded as well when  ph is bounded, which is not the case. The same remark holds with  E. ,  E! ,  E*. (Contributed by BJ, 16-Oct-2019.)

 |- BOUNDED  ph   =>    |- BOUNDED  E! x  e.  y  ph
 
Theorembdrmo 10932* Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  ph   =>    |- BOUNDED  E* x  e.  y  ph
 
Theorembdcvv 10933 The universal class is bounded. The formulation may sound strange, but recall that here, "bounded" means "Δ0". (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  _V
 
Theorembdsbc 10934 A formula resulting from proper substitution of a setvar for a setvar in a bounded formula is bounded. See also bdsbcALT 10935. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  ph   =>    |- BOUNDED  [. y  /  x ]. ph
 
TheorembdsbcALT 10935 Alternate proof of bdsbc 10934. (Contributed by BJ, 16-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- BOUNDED  ph   =>    |- BOUNDED  [. y  /  x ]. ph
 
Theorembdccsb 10936 A class resulting from proper substitution of a setvar for a setvar in a bounded class is bounded. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  A   =>    |- BOUNDED  [_ y  /  x ]_ A
 
Theorembdcdif 10937 The difference of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  A   &    |- BOUNDED  B   =>    |- BOUNDED  ( A 
 \  B )
 
Theorembdcun 10938 The union of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  A   &    |- BOUNDED  B   =>    |- BOUNDED  ( A  u.  B )
 
Theorembdcin 10939 The intersection of two bounded classes is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  A   &    |- BOUNDED  B   =>    |- BOUNDED  ( A  i^i  B )
 
Theorembdss 10940 The inclusion of a setvar in a bounded class is a bounded formula. Note: apparently, we cannot prove from the present axioms that equality of two bounded classes is a bounded formula. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  A   =>    |- BOUNDED  x  C_  A
 
Theorembdcnul 10941 The empty class is bounded. See also bdcnulALT 10942. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  (/)
 
TheorembdcnulALT 10942 Alternate proof of bdcnul 10941. Similarly, for the next few theorems proving boundedness of a class, one can either use their definition followed by bdceqir 10920, or use the corresponding characterizations of its elements followed by bdelir 10923. (Contributed by BJ, 3-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- BOUNDED  (/)
 
Theorembdeq0 10943 Boundedness of the formula expressing that a setvar is equal to the empty class. (Contributed by BJ, 21-Nov-2019.)
 |- BOUNDED  x  =  (/)
 
Theorembj-bd0el 10944 Boundedness of the formula "the empty set belongs to the setvar  x". (Contributed by BJ, 30-Nov-2019.)
 |- BOUNDED  (/)  e.  x
 
Theorembdcpw 10945 The power class of a bounded class is bounded. (Contributed by BJ, 3-Oct-2019.)
 |- BOUNDED  A   =>    |- BOUNDED  ~P A
 
Theorembdcsn 10946 The singleton of a setvar is bounded. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  { x }
 
Theorembdcpr 10947 The pair of two setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  { x ,  y }
 
Theorembdctp 10948 The unordered triple of three setvars is bounded. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  { x ,  y ,  z }
 
Theorembdsnss 10949* Inclusion of a singleton of a setvar in a bounded class is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  A   =>    |- BOUNDED  { x }  C_  A
 
Theorembdvsn 10950* Equality of a setvar with a singleton of a setvar is a bounded formula. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  x  =  { y }
 
Theorembdop 10951 The ordered pair of two setvars is a bounded class. (Contributed by BJ, 21-Nov-2019.)
 |- BOUNDED 
 <. x ,  y >.
 
Theorembdcuni 10952 The union of a setvar is a bounded class. (Contributed by BJ, 15-Oct-2019.)
 |- BOUNDED 
 U. x
 
Theorembdcint 10953 The intersection of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED 
 |^| x
 
Theorembdciun 10954* The indexed union of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  A   =>    |- BOUNDED  U_ x  e.  y  A
 
Theorembdciin 10955* The indexed intersection of a bounded class with a setvar indexing set is a bounded class. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED  A   =>    |- BOUNDED  |^|_ x  e.  y  A
 
Theorembdcsuc 10956 The successor of a setvar is a bounded class. (Contributed by BJ, 16-Oct-2019.)
 |- BOUNDED 
 suc  x
 
Theorembdeqsuc 10957* Boundedness of the formula expressing that a setvar is equal to the successor of another. (Contributed by BJ, 21-Nov-2019.)
 |- BOUNDED  x  =  suc  y
 
Theorembj-bdsucel 10958 Boundedness of the formula "the successor of the setvar  x belongs to the setvar  y". (Contributed by BJ, 30-Nov-2019.)
 |- BOUNDED  suc  x  e.  y
 
Theorembdcriota 10959* A class given by a restricted definition binder is bounded, under the given hypotheses. (Contributed by BJ, 24-Nov-2019.)
 |- BOUNDED  ph   &    |-  E! x  e.  y  ph   =>    |- BOUNDED  ( iota_ x  e.  y  ph )
 
6.2.6  CZF: Bounded separation

In this section, we state the axiom scheme of bounded separation, which is part of CZF set theory.

 
Axiomax-bdsep 10960* Axiom scheme of bounded (or restricted, or Δ0) separation. It is stated with all possible disjoint variable conditions, to show that this weak form is sufficient. For the full axiom of separation, see ax-sep 3916. (Contributed by BJ, 5-Oct-2019.)
 |- BOUNDED  ph   =>    |- 
 A. a E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
 )
 
Theorembdsep1 10961* Version of ax-bdsep 10960 without initial universal quantifier. (Contributed by BJ, 5-Oct-2019.)
 |- BOUNDED  ph   =>    |- 
 E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph ) )
 
Theorembdsep2 10962* Version of ax-bdsep 10960 with one DV condition removed and without initial universal quantifier. Use bdsep1 10961 when sufficient. (Contributed by BJ, 5-Oct-2019.)
 |- BOUNDED  ph   =>    |- 
 E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph ) )
 
Theorembdsepnft 10963* Closed form of bdsepnf 10964. Version of ax-bdsep 10960 with one DV condition removed, the other DV condition replaced by a non-freeness antecedent, and without initial universal quantifier. Use bdsep1 10961 when sufficient. (Contributed by BJ, 19-Oct-2019.)
 |- BOUNDED  ph   =>    |-  ( A. x F/ b ph  ->  E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph )
 ) )
 
Theorembdsepnf 10964* Version of ax-bdsep 10960 with one DV condition removed, the other DV condition replaced by a non-freeness hypothesis, and without initial universal quantifier. See also bdsepnfALT 10965. Use bdsep1 10961 when sufficient. (Contributed by BJ, 5-Oct-2019.)
 |-  F/ b ph   &    |- BOUNDED  ph   =>    |- 
 E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph ) )
 
TheorembdsepnfALT 10965* Alternate proof of bdsepnf 10964, not using bdsepnft 10963. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  F/ b ph   &    |- BOUNDED  ph   =>    |- 
 E. b A. x ( x  e.  b  <->  ( x  e.  a  /\  ph ) )
 
Theorembdzfauscl 10966* Closed form of the version of zfauscl 3918 for bounded formulas using bounded separation. (Contributed by BJ, 13-Nov-2019.)
 |- BOUNDED  ph   =>    |-  ( A  e.  V  ->  E. y A. x ( x  e.  y  <->  ( x  e.  A  /\  ph ) ) )
 
Theorembdbm1.3ii 10967* Bounded version of bm1.3ii 3919. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |- BOUNDED  ph   &    |-  E. x A. y ( ph  ->  y  e.  x )   =>    |-  E. x A. y ( y  e.  x  <->  ph )
 
Theorembj-axemptylem 10968* Lemma for bj-axempty 10969 and bj-axempty2 10970. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3924 instead. (New usage is discouraged.)
 |-  E. x A. y ( y  e.  x  -> F.  )
 
Theorembj-axempty 10969* Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a non-empty universe. See axnul 3923. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3924 instead. (New usage is discouraged.)
 |-  E. x A. y  e.  x F.
 
Theorembj-axempty2 10970* Axiom of the empty set from bounded separation, alternate version to bj-axempty 10969. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3924 instead. (New usage is discouraged.)
 |-  E. x A. y  -.  y  e.  x
 
Theorembj-nalset 10971* nalset 3928 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  -.  E. x A. y  y  e.  x
 
Theorembj-vprc 10972 vprc 3929 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  -.  _V  e.  _V
 
Theorembj-nvel 10973 nvel 3930 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  -.  _V  e.  A
 
Theorembj-vnex 10974 vnex 3931 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  -.  E. x  x  =  _V
 
Theorembdinex1 10975 Bounded version of inex1 3932. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  B   &    |-  A  e.  _V   =>    |-  ( A  i^i  B )  e. 
 _V
 
Theorembdinex2 10976 Bounded version of inex2 3933. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  B   &    |-  A  e.  _V   =>    |-  ( B  i^i  A )  e. 
 _V
 
Theorembdinex1g 10977 Bounded version of inex1g 3934. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  B   =>    |-  ( A  e.  V  ->  ( A  i^i  B )  e.  _V )
 
Theorembdssex 10978 Bounded version of ssex 3935. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   &    |-  B  e.  _V   =>    |-  ( A  C_  B  ->  A  e.  _V )
 
Theorembdssexi 10979 Bounded version of ssexi 3936. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   &    |-  B  e.  _V   &    |-  A  C_  B   =>    |-  A  e.  _V
 
Theorembdssexg 10980 Bounded version of ssexg 3937. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   =>    |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  e.  _V )
 
Theorembdssexd 10981 Bounded version of ssexd 3938. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  B  e.  C )   &    |-  ( ph  ->  A  C_  B )   &    |- BOUNDED  A   =>    |-  ( ph  ->  A  e.  _V )
 
Theorembdrabexg 10982* Bounded version of rabexg 3941. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  ph   &    |- BOUNDED  A   =>    |-  ( A  e.  V  ->  { x  e.  A  |  ph }  e.  _V )
 
Theorembj-inex 10983 The intersection of two sets is a set, from bounded separation. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A  i^i  B )  e.  _V )
 
Theorembj-intexr 10984 intexr 3945 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  ( |^| A  e.  _V  ->  A  =/=  (/) )
 
Theorembj-intnexr 10985 intnexr 3946 from bounded separation. (Contributed by BJ, 18-Nov-2019.) (Proof modification is discouraged.)
 |-  ( |^| A  =  _V  ->  -. 
 |^| A  e.  _V )
 
Theorembj-zfpair2 10986 Proof of zfpair2 3993 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |-  { x ,  y }  e.  _V
 
Theorembj-prexg 10987 Proof of prexg 3994 using only bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  { A ,  B }  e.  _V )
 
Theorembj-snexg 10988 snexg 3976 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  { A }  e.  _V )
 
Theorembj-snex 10989 snex 3977 from bounded separation. (Contributed by BJ, 5-Oct-2019.) (Proof modification is discouraged.)
 |-  A  e.  _V   =>    |- 
 { A }  e.  _V
 
Theorembj-sels 10990* If a class is a set, then it is a member of a set. (Copied from set.mm.) (Contributed by BJ, 3-Apr-2019.)
 |-  ( A  e.  V  ->  E. x  A  e.  x )
 
Theorembj-axun2 10991* axun2 4218 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
 |-  E. y A. z ( z  e.  y  <->  E. w ( z  e.  w  /\  w  e.  x ) )
 
Theorembj-uniex2 10992* uniex2 4219 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)
 |-  E. y  y  =  U. x
 
Theorembj-uniex 10993 uniex 4220 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  A  e.  _V   =>    |- 
 U. A  e.  _V
 
Theorembj-uniexg 10994 uniexg 4221 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  U. A  e.  _V )
 
Theorembj-unex 10995 unex 4222 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  u.  B )  e. 
 _V
 
Theorembdunexb 10996 Bounded version of unexb 4223. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |- BOUNDED  A   &    |- BOUNDED  B   =>    |-  ( ( A  e.  _V 
 /\  B  e.  _V ) 
 <->  ( A  u.  B )  e.  _V )
 
Theorembj-unexg 10997 unexg 4224 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A  u.  B )  e.  _V )
 
Theorembj-sucexg 10998 sucexg 4270 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  ( A  e.  V  ->  suc 
 A  e.  _V )
 
Theorembj-sucex 10999 sucex 4271 from bounded separation. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
 |-  A  e.  _V   =>    |- 
 suc  A  e.  _V
 
6.2.6.1  Delta_0-classical logic
 
Axiomax-bj-d0cl 11000 Axiom for Δ0-classical logic. (Contributed by BJ, 2-Jan-2020.)
 |- BOUNDED  ph   =>    |- DECID  ph
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