Theorem tz6.26

Description: All nonempty subclasses of a class having a well-ordered set-like relation have minimal elements for that relation. Proposition 6.26 of TakeutiZaring p. 31. (Contributed by Scott Fenton, 29-Jan-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Assertion	tz6.26	⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐵 , 𝑦 ) = ∅ )

Proof

Step	Hyp	Ref	Expression
1		wereu2	⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃! 𝑦 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 )
2		reurex	⊢ ( ∃! 𝑦 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 → ∃ 𝑦 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 )
3	1 2	syl	⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑦 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 )
4		rabeq0	⊢ ( { 𝑥 ∈ 𝐵 ∣ 𝑥 𝑅 𝑦 } = ∅ ↔ ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 )
5		dfrab3	⊢ { 𝑥 ∈ 𝐵 ∣ 𝑥 𝑅 𝑦 } = ( 𝐵 ∩ { 𝑥 ∣ 𝑥 𝑅 𝑦 } )
6		vex	⊢ 𝑦 ∈ V
7	6	dfpred2	⊢ Pred ( 𝑅 , 𝐵 , 𝑦 ) = ( 𝐵 ∩ { 𝑥 ∣ 𝑥 𝑅 𝑦 } )
8	5 7	eqtr4i	⊢ { 𝑥 ∈ 𝐵 ∣ 𝑥 𝑅 𝑦 } = Pred ( 𝑅 , 𝐵 , 𝑦 )
9	8	eqeq1i	⊢ ( { 𝑥 ∈ 𝐵 ∣ 𝑥 𝑅 𝑦 } = ∅ ↔ Pred ( 𝑅 , 𝐵 , 𝑦 ) = ∅ )
10	4 9	bitr3i	⊢ ( ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ↔ Pred ( 𝑅 , 𝐵 , 𝑦 ) = ∅ )
11	10	rexbii	⊢ ( ∃ 𝑦 ∈ 𝐵 ∀ 𝑥 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ↔ ∃ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐵 , 𝑦 ) = ∅ )
12	3 11	sylib	⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑦 ∈ 𝐵 Pred ( 𝑅 , 𝐵 , 𝑦 ) = ∅ )