Theorem bnj125

Description: Technical lemma for bnj150 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj125.1	$$⊢\phi \leftrightarrow f\left(\varnothing \right)=pred\left(x,A,R\right)$$
	bnj125.2	No typesetting found for |- ( ph' <-> [. 1o / n ]. ph ) with typecode |-
	bnj125.3	No typesetting found for |- ( ph" <-> [. F / f ]. ph' ) with typecode |-
	bnj125.4	$$⊢F=\left\{⟨\varnothing ,pred\left(x,A,R\right)⟩\right\}$$
Assertion	bnj125	Could not format assertion : No typesetting found for |- ( ph" <-> ( F ` (/) ) = _pred ( x , A , R ) ) with typecode |-

Proof

Step	Hyp	Ref	Expression
1		bnj125.1	$$⊢\phi \leftrightarrow f\left(\varnothing \right)=pred\left(x,A,R\right)$$
2		bnj125.2	Could not format ( ph' <-> [. 1o / n ]. ph ) : No typesetting found for |- ( ph' <-> [. 1o / n ]. ph ) with typecode |-
3		bnj125.3	Could not format ( ph" <-> [. F / f ]. ph' ) : No typesetting found for |- ( ph" <-> [. F / f ]. ph' ) with typecode |-
4		bnj125.4	$$⊢F=\left\{⟨\varnothing ,pred\left(x,A,R\right)⟩\right\}$$
5	2	sbcbii	Could not format ( [. F / f ]. ph' <-> [. F / f ]. [. 1o / n ]. ph ) : No typesetting found for |- ( [. F / f ]. ph' <-> [. F / f ]. [. 1o / n ]. ph ) with typecode |-
6		bnj105	$$⊢1_{𝑜}\in \mathrm{V}$$
7	1 6	bnj91	$$⊢\underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\phi \leftrightarrow f\left(\varnothing \right)=pred\left(x,A,R\right)$$
8	7	sbcbii	$$⊢\underset{\dot{}}{[}F/f\underset{\dot{}}{]}\underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\phi \leftrightarrow \underset{\dot{}}{[}F/f\underset{\dot{}}{]}f\left(\varnothing \right)=pred\left(x,A,R\right)$$
9	4	bnj95	$$⊢F\in \mathrm{V}$$
10		fveq1	$$⊢f=F\to f\left(\varnothing \right)=F\left(\varnothing \right)$$
11	10	eqeq1d	$$⊢f=F\to \left(f\left(\varnothing \right)=pred\left(x,A,R\right)\leftrightarrow F\left(\varnothing \right)=pred\left(x,A,R\right)\right)$$
12	9 11	sbcie	$$⊢\underset{\dot{}}{[}F/f\underset{\dot{}}{]}f\left(\varnothing \right)=pred\left(x,A,R\right)\leftrightarrow F\left(\varnothing \right)=pred\left(x,A,R\right)$$
13	8 12	bitri	$$⊢\underset{\dot{}}{[}F/f\underset{\dot{}}{]}\underset{\dot{}}{[}{1}_{𝑜}/n\underset{\dot{}}{]}\phi \leftrightarrow F\left(\varnothing \right)=pred\left(x,A,R\right)$$
14	5 13	bitri	Could not format ( [. F / f ]. ph' <-> ( F ` (/) ) = _pred ( x , A , R ) ) : No typesetting found for |- ( [. F / f ]. ph' <-> ( F ` (/) ) = _pred ( x , A , R ) ) with typecode |-
15	3 14	bitri	Could not format ( ph" <-> ( F ` (/) ) = _pred ( x , A , R ) ) : No typesetting found for |- ( ph" <-> ( F ` (/) ) = _pred ( x , A , R ) ) with typecode |-