Theorem upgr1elem

Description: Lemma for upgr1e and uspgr1e . (Contributed by AV, 16-Oct-2020)

	Ref	Expression
Hypotheses	upgr1elem.s	$$⊢\phi \to \left\{B,C\right\}\in S$$
	upgr1elem.b	$$⊢\phi \to B\in W$$
Assertion	upgr1elem	$$⊢\phi \to \left\{\left\{B,C\right\}\right\}\subseteq \left\{x\in \left(S\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}$$

Proof

Step	Hyp	Ref	Expression
1		upgr1elem.s	$$⊢\phi \to \left\{B,C\right\}\in S$$
2		upgr1elem.b	$$⊢\phi \to B\in W$$
3		fveq2	$$⊢x=\left\{B,C\right\}\to \left|x\right|=\left|\left\{B,C\right\}\right|$$
4	3	breq1d	$$⊢x=\left\{B,C\right\}\to \left(\left|x\right|\le 2\leftrightarrow \left|\left\{B,C\right\}\right|\le 2\right)$$
5		prnzg	$$⊢B\in W\to \left\{B,C\right\}\ne \varnothing$$
6	2 5	syl	$$⊢\phi \to \left\{B,C\right\}\ne \varnothing$$
7		eldifsn	$$⊢\left\{B,C\right\}\in \left(S\setminus \left\{\varnothing \right\}\right)\leftrightarrow \left(\left\{B,C\right\}\in S\wedge \left\{B,C\right\}\ne \varnothing \right)$$
8	1 6 7	sylanbrc	$$⊢\phi \to \left\{B,C\right\}\in \left(S\setminus \left\{\varnothing \right\}\right)$$
9		hashprlei	$$⊢\left(\left\{B,C\right\}\in \mathrm{Fin}\wedge \left|\left\{B,C\right\}\right|\le 2\right)$$
10	9	simpri	$$⊢\left|\left\{B,C\right\}\right|\le 2$$
11	10	a1i	$$⊢\phi \to \left|\left\{B,C\right\}\right|\le 2$$
12	4 8 11	elrabd	$$⊢\phi \to \left\{B,C\right\}\in \left\{x\in \left(S\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}$$
13	12	snssd	$$⊢\phi \to \left\{\left\{B,C\right\}\right\}\subseteq \left\{x\in \left(S\setminus \left\{\varnothing \right\}\right)|\left|x\right|\le 2\right\}$$