Theorem htaOLD

Description: Obsolete version of hta as of 22-Jul-2026. (Contributed by NM, 11-Mar-2004) (Revised by Mario Carneiro, 25-Jun-2015) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	htaOLD.1	$$⊢A=\left\{x|\left(\phi \wedge \forall y\left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \to \mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right)\right)\right\}$$
	htaOLD.2	$$⊢B=\left(\iota z\in A|\forall w\in A\neg wRz\right)$$
Assertion	htaOLD	$$⊢R\mathrm{We}A\to \left(\phi \to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		htaOLD.1	$$⊢A=\left\{x|\left(\phi \wedge \forall y\left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \to \mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right)\right)\right\}$$
2		htaOLD.2	$$⊢B=\left(\iota z\in A|\forall w\in A\neg wRz\right)$$
3		19.8a	$$⊢\phi \to \exists x\phi$$
4		scott0bsOLD	$$⊢\exists x\phi \leftrightarrow \left\{x|\left(\phi \wedge \forall y\left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \to \mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right)\right)\right\}\ne \varnothing$$
5	1	neeq1i	$$⊢A\ne \varnothing \leftrightarrow \left\{x|\left(\phi \wedge \forall y\left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \to \mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right)\right)\right\}\ne \varnothing$$
6	4 5	bitr4i	$$⊢\exists x\phi \leftrightarrow A\ne \varnothing$$
7	3 6	sylib	$$⊢\phi \to A\ne \varnothing$$
8		scottexsOLD	$$⊢\left\{x|\left(\phi \wedge \forall y\left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \to \mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right)\right)\right\}\in \mathrm{V}$$
9	1 8	eqeltri	$$⊢A\in \mathrm{V}$$
10	9 2	htalem	$$⊢\left(R\mathrm{We}A\wedge A\ne \varnothing \right)\to B\in A$$
11	10	ex	$$⊢R\mathrm{We}A\to \left(A\ne \varnothing \to B\in A\right)$$
12		simpl	$$⊢\left(\phi \wedge \forall y\left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \to \mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right)\right)\to \phi$$
13	12	ss2abi	$$⊢\left\{x|\left(\phi \wedge \forall y\left(\underset{\dot{}}{[}y/x\underset{\dot{}}{]}\phi \to \mathrm{rank}\left(x\right)\subseteq \mathrm{rank}\left(y\right)\right)\right)\right\}\subseteq \left\{x|\phi \right\}$$
14	1 13	eqsstri	$$⊢A\subseteq \left\{x|\phi \right\}$$
15	14	sseli	$$⊢B\in A\to B\in \left\{x|\phi \right\}$$
16		df-sbc	$$⊢\underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \leftrightarrow B\in \left\{x|\phi \right\}$$
17	15 16	sylibr	$$⊢B\in A\to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi$$
18	7 11 17	syl56	$$⊢R\mathrm{We}A\to \left(\phi \to \underset{\dot{}}{[}B/x\underset{\dot{}}{]}\phi \right)$$