Metamath Proof Explorer

Theorem wefrc

Description: A nonempty subclass of a class well-ordered by membership has a minimal element. Special case of Proposition 6.26 of TakeutiZaring p. 31. (Contributed by NM, 17-Feb-2004)

Ref Expression
Assertion wefrc ( ( E We 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐵 ( 𝐵 ∩ 𝑥 ) = ∅ )

Proof

Step Hyp Ref Expression
1 wess ⊢ ( 𝐵 ⊆ 𝐴 → ( E We 𝐴 → E We 𝐵 ) )
2 n0 ⊢ ( 𝐵 ≠ ∅ ↔ ∃ 𝑦 𝑦 ∈ 𝐵 )
3 ineq2 ⊢ ( 𝑥 = 𝑦 → ( 𝐵 ∩ 𝑥 ) = ( 𝐵 ∩ 𝑦 ) )
4 3 eqeq1d ⊢ ( 𝑥 = 𝑦 → ( ( 𝐵 ∩ 𝑥 ) = ∅ ↔ ( 𝐵 ∩ 𝑦 ) = ∅ ) )
5 4 rspcev ⊢ ( ( 𝑦 ∈ 𝐵 ∧ ( 𝐵 ∩ 𝑦 ) = ∅ ) → ∃ 𝑥 ∈ 𝐵 ( 𝐵 ∩ 𝑥 ) = ∅ )
6 5 ex ⊢ ( 𝑦 ∈ 𝐵 → ( ( 𝐵 ∩ 𝑦 ) = ∅ → ∃ 𝑥 ∈ 𝐵 ( 𝐵 ∩ 𝑥 ) = ∅ ) )
7 6 adantl ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝐵 ∩ 𝑦 ) = ∅ → ∃ 𝑥 ∈ 𝐵 ( 𝐵 ∩ 𝑥 ) = ∅ ) )
8 inss1 ⊢ ( 𝐵 ∩ 𝑦 ) ⊆ 𝐵
9 wefr ⊢ ( E We 𝐵 → E Fr 𝐵 )
10 vex ⊢ 𝑦 ∈ V
11 10 inex2 ⊢ ( 𝐵 ∩ 𝑦 ) ∈ V
12 11 epfrc ⊢ ( ( E Fr 𝐵 ∧ ( 𝐵 ∩ 𝑦 ) ⊆ 𝐵 ∧ ( 𝐵 ∩ 𝑦 ) ≠ ∅ ) → ∃ 𝑥 ∈ ( 𝐵 ∩ 𝑦 ) ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ )
13 9 12 syl3an1 ⊢ ( ( E We 𝐵 ∧ ( 𝐵 ∩ 𝑦 ) ⊆ 𝐵 ∧ ( 𝐵 ∩ 𝑦 ) ≠ ∅ ) → ∃ 𝑥 ∈ ( 𝐵 ∩ 𝑦 ) ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ )
14 13 3exp ⊢ ( E We 𝐵 → ( ( 𝐵 ∩ 𝑦 ) ⊆ 𝐵 → ( ( 𝐵 ∩ 𝑦 ) ≠ ∅ → ∃ 𝑥 ∈ ( 𝐵 ∩ 𝑦 ) ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ ) ) )
15 8 14 mpi ⊢ ( E We 𝐵 → ( ( 𝐵 ∩ 𝑦 ) ≠ ∅ → ∃ 𝑥 ∈ ( 𝐵 ∩ 𝑦 ) ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ ) )
16 rexin ⊢ ( ∃ 𝑥 ∈ ( 𝐵 ∩ 𝑦 ) ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ ↔ ∃ 𝑥 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ ) )
17 15 16 imbitrdi ⊢ ( E We 𝐵 → ( ( 𝐵 ∩ 𝑦 ) ≠ ∅ → ∃ 𝑥 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ ) ) )
18 17 adantr ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝐵 ∩ 𝑦 ) ≠ ∅ → ∃ 𝑥 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ ) ) )
19 elin ⊢ ( 𝑧 ∈ ( 𝐵 ∩ 𝑥 ) ↔ ( 𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝑥 ) )
20 df-3an ⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ↔ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) )
21 3anrot ⊢ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ) ↔ ( 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) )
22 20 21 bitr3i ⊢ ( ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ↔ ( 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) )
23 wetrep ⊢ ( ( E We 𝐵 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝑦 ) → 𝑧 ∈ 𝑦 ) )
24 23 expd ⊢ ( ( E We 𝐵 ∧ ( 𝑧 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑧 ∈ 𝑥 → ( 𝑥 ∈ 𝑦 → 𝑧 ∈ 𝑦 ) ) )
25 22 24 sylan2b ⊢ ( ( E We 𝐵 ∧ ( ( 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵 ) ∧ 𝑥 ∈ 𝐵 ) ) → ( 𝑧 ∈ 𝑥 → ( 𝑥 ∈ 𝑦 → 𝑧 ∈ 𝑦 ) ) )
26 25 exp44 ⊢ ( E We 𝐵 → ( 𝑦 ∈ 𝐵 → ( 𝑧 ∈ 𝐵 → ( 𝑥 ∈ 𝐵 → ( 𝑧 ∈ 𝑥 → ( 𝑥 ∈ 𝑦 → 𝑧 ∈ 𝑦 ) ) ) ) ) )
27 26 imp ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑧 ∈ 𝐵 → ( 𝑥 ∈ 𝐵 → ( 𝑧 ∈ 𝑥 → ( 𝑥 ∈ 𝑦 → 𝑧 ∈ 𝑦 ) ) ) ) )
28 27 com34 ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑧 ∈ 𝐵 → ( 𝑧 ∈ 𝑥 → ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝑦 → 𝑧 ∈ 𝑦 ) ) ) ) )
29 28 impd ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑧 ∈ 𝐵 ∧ 𝑧 ∈ 𝑥 ) → ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝑦 → 𝑧 ∈ 𝑦 ) ) ) )
30 19 29 biimtrid ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑧 ∈ ( 𝐵 ∩ 𝑥 ) → ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝑦 → 𝑧 ∈ 𝑦 ) ) ) )
31 30 imp4a ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑧 ∈ ( 𝐵 ∩ 𝑥 ) → ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝑦 ) → 𝑧 ∈ 𝑦 ) ) )
32 31 com23 ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝑦 ) → ( 𝑧 ∈ ( 𝐵 ∩ 𝑥 ) → 𝑧 ∈ 𝑦 ) ) )
33 32 ralrimdv ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝑦 ) → ∀ 𝑧 ∈ ( 𝐵 ∩ 𝑥 ) 𝑧 ∈ 𝑦 ) )
34 dfss3 ⊢ ( ( 𝐵 ∩ 𝑥 ) ⊆ 𝑦 ↔ ∀ 𝑧 ∈ ( 𝐵 ∩ 𝑥 ) 𝑧 ∈ 𝑦 )
35 33 34 imbitrrdi ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝑦 ) → ( 𝐵 ∩ 𝑥 ) ⊆ 𝑦 ) )
36 dfss ⊢ ( ( 𝐵 ∩ 𝑥 ) ⊆ 𝑦 ↔ ( 𝐵 ∩ 𝑥 ) = ( ( 𝐵 ∩ 𝑥 ) ∩ 𝑦 ) )
37 in32 ⊢ ( ( 𝐵 ∩ 𝑥 ) ∩ 𝑦 ) = ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 )
38 37 eqeq2i ⊢ ( ( 𝐵 ∩ 𝑥 ) = ( ( 𝐵 ∩ 𝑥 ) ∩ 𝑦 ) ↔ ( 𝐵 ∩ 𝑥 ) = ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) )
39 36 38 sylbb ⊢ ( ( 𝐵 ∩ 𝑥 ) ⊆ 𝑦 → ( 𝐵 ∩ 𝑥 ) = ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) )
40 39 eqeq1d ⊢ ( ( 𝐵 ∩ 𝑥 ) ⊆ 𝑦 → ( ( 𝐵 ∩ 𝑥 ) = ∅ ↔ ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ ) )
41 40 biimprd ⊢ ( ( 𝐵 ∩ 𝑥 ) ⊆ 𝑦 → ( ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ → ( 𝐵 ∩ 𝑥 ) = ∅ ) )
42 35 41 syl6 ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝑦 ) → ( ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ → ( 𝐵 ∩ 𝑥 ) = ∅ ) ) )
43 42 expd ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐵 → ( 𝑥 ∈ 𝑦 → ( ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ → ( 𝐵 ∩ 𝑥 ) = ∅ ) ) ) )
44 43 imp4a ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( 𝑥 ∈ 𝐵 → ( ( 𝑥 ∈ 𝑦 ∧ ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ ) → ( 𝐵 ∩ 𝑥 ) = ∅ ) ) )
45 44 reximdvai ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ∃ 𝑥 ∈ 𝐵 ( 𝑥 ∈ 𝑦 ∧ ( ( 𝐵 ∩ 𝑦 ) ∩ 𝑥 ) = ∅ ) → ∃ 𝑥 ∈ 𝐵 ( 𝐵 ∩ 𝑥 ) = ∅ ) )
46 18 45 syld ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ( ( 𝐵 ∩ 𝑦 ) ≠ ∅ → ∃ 𝑥 ∈ 𝐵 ( 𝐵 ∩ 𝑥 ) = ∅ ) )
47 7 46 pm2.61dne ⊢ ( ( E We 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ∃ 𝑥 ∈ 𝐵 ( 𝐵 ∩ 𝑥 ) = ∅ )
48 47 ex ⊢ ( E We 𝐵 → ( 𝑦 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐵 ( 𝐵 ∩ 𝑥 ) = ∅ ) )
49 48 exlimdv ⊢ ( E We 𝐵 → ( ∃ 𝑦 𝑦 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐵 ( 𝐵 ∩ 𝑥 ) = ∅ ) )
50 2 49 biimtrid ⊢ ( E We 𝐵 → ( 𝐵 ≠ ∅ → ∃ 𝑥 ∈ 𝐵 ( 𝐵 ∩ 𝑥 ) = ∅ ) )
51 1 50 syl6com ⊢ ( E We 𝐴 → ( 𝐵 ⊆ 𝐴 → ( 𝐵 ≠ ∅ → ∃ 𝑥 ∈ 𝐵 ( 𝐵 ∩ 𝑥 ) = ∅ ) ) )
52 51 3imp ⊢ ( ( E We 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) → ∃ 𝑥 ∈ 𝐵 ( 𝐵 ∩ 𝑥 ) = ∅ )