Metamath Proof Explorer

Theorem php

Description: Pigeonhole Principle. A natural number is not equinumerous to a proper subset of itself. Theorem (Pigeonhole Principle) of Enderton p. 134. The theorem is so-called because you can't putn + 1 pigeons inton holes (if each hole holds only one pigeon). The proof consists of lemmas phplem1 through phplem4 , nneneq , and this final piece of the proof. (Contributed by NM, 29-May-1998)

Ref Expression
Assertion php ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → ¬ 𝐴 ≈ 𝐵 )

Proof

Step Hyp Ref Expression
1 0ss ⊢ ∅ ⊆ 𝐵
2 sspsstr ⊢ ( ( ∅ ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐴 ) → ∅ ⊊ 𝐴 )
3 1 2 mpan ⊢ ( 𝐵 ⊊ 𝐴 → ∅ ⊊ 𝐴 )
4 0pss ⊢ ( ∅ ⊊ 𝐴 ↔ 𝐴 ≠ ∅ )
5 df-ne ⊢ ( 𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅ )
6 4 5 bitri ⊢ ( ∅ ⊊ 𝐴 ↔ ¬ 𝐴 = ∅ )
7 3 6 sylib ⊢ ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 = ∅ )
8 nn0suc ⊢ ( 𝐴 ∈ ω → ( 𝐴 = ∅ ∨ ∃ 𝑥 ∈ ω 𝐴 = suc 𝑥 ) )
9 8 orcanai ⊢ ( ( 𝐴 ∈ ω ∧ ¬ 𝐴 = ∅ ) → ∃ 𝑥 ∈ ω 𝐴 = suc 𝑥 )
10 7 9 sylan2 ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → ∃ 𝑥 ∈ ω 𝐴 = suc 𝑥 )
11 pssnel ⊢ ( 𝐵 ⊊ suc 𝑥 → ∃ 𝑦 ( 𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵 ) )
12 pssss ⊢ ( 𝐵 ⊊ suc 𝑥 → 𝐵 ⊆ suc 𝑥 )
13 ssdif ⊢ ( 𝐵 ⊆ suc 𝑥 → ( 𝐵 ∖ { 𝑦 } ) ⊆ ( suc 𝑥 ∖ { 𝑦 } ) )
14 disjsn ⊢ ( ( 𝐵 ∩ { 𝑦 } ) = ∅ ↔ ¬ 𝑦 ∈ 𝐵 )
15 disj3 ⊢ ( ( 𝐵 ∩ { 𝑦 } ) = ∅ ↔ 𝐵 = ( 𝐵 ∖ { 𝑦 } ) )
16 14 15 bitr3i ⊢ ( ¬ 𝑦 ∈ 𝐵 ↔ 𝐵 = ( 𝐵 ∖ { 𝑦 } ) )
17 sseq1 ⊢ ( 𝐵 = ( 𝐵 ∖ { 𝑦 } ) → ( 𝐵 ⊆ ( suc 𝑥 ∖ { 𝑦 } ) ↔ ( 𝐵 ∖ { 𝑦 } ) ⊆ ( suc 𝑥 ∖ { 𝑦 } ) ) )
18 16 17 sylbi ⊢ ( ¬ 𝑦 ∈ 𝐵 → ( 𝐵 ⊆ ( suc 𝑥 ∖ { 𝑦 } ) ↔ ( 𝐵 ∖ { 𝑦 } ) ⊆ ( suc 𝑥 ∖ { 𝑦 } ) ) )
19 13 18 syl5ibr ⊢ ( ¬ 𝑦 ∈ 𝐵 → ( 𝐵 ⊆ suc 𝑥 → 𝐵 ⊆ ( suc 𝑥 ∖ { 𝑦 } ) ) )
20 vex ⊢ 𝑥 ∈ V
21 20 sucex ⊢ suc 𝑥 ∈ V
22 21 difexi ⊢ ( suc 𝑥 ∖ { 𝑦 } ) ∈ V
23 ssdomg ⊢ ( ( suc 𝑥 ∖ { 𝑦 } ) ∈ V → ( 𝐵 ⊆ ( suc 𝑥 ∖ { 𝑦 } ) → 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ) )
24 22 23 ax-mp ⊢ ( 𝐵 ⊆ ( suc 𝑥 ∖ { 𝑦 } ) → 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) )
25 12 19 24 syl56 ⊢ ( ¬ 𝑦 ∈ 𝐵 → ( 𝐵 ⊊ suc 𝑥 → 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ) )
26 25 imp ⊢ ( ( ¬ 𝑦 ∈ 𝐵 ∧ 𝐵 ⊊ suc 𝑥 ) → 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) )
27 vex ⊢ 𝑦 ∈ V
28 20 27 phplem3 ⊢ ( ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) → 𝑥 ≈ ( suc 𝑥 ∖ { 𝑦 } ) )
29 28 ensymd ⊢ ( ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) → ( suc 𝑥 ∖ { 𝑦 } ) ≈ 𝑥 )
30 domentr ⊢ ( ( 𝐵 ≼ ( suc 𝑥 ∖ { 𝑦 } ) ∧ ( suc 𝑥 ∖ { 𝑦 } ) ≈ 𝑥 ) → 𝐵 ≼ 𝑥 )
31 26 29 30 syl2an ⊢ ( ( ( ¬ 𝑦 ∈ 𝐵 ∧ 𝐵 ⊊ suc 𝑥 ) ∧ ( 𝑥 ∈ ω ∧ 𝑦 ∈ suc 𝑥 ) ) → 𝐵 ≼ 𝑥 )
32 31 exp43 ⊢ ( ¬ 𝑦 ∈ 𝐵 → ( 𝐵 ⊊ suc 𝑥 → ( 𝑥 ∈ ω → ( 𝑦 ∈ suc 𝑥 → 𝐵 ≼ 𝑥 ) ) ) )
33 32 com4r ⊢ ( 𝑦 ∈ suc 𝑥 → ( ¬ 𝑦 ∈ 𝐵 → ( 𝐵 ⊊ suc 𝑥 → ( 𝑥 ∈ ω → 𝐵 ≼ 𝑥 ) ) ) )
34 33 imp ⊢ ( ( 𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵 ) → ( 𝐵 ⊊ suc 𝑥 → ( 𝑥 ∈ ω → 𝐵 ≼ 𝑥 ) ) )
35 34 exlimiv ⊢ ( ∃ 𝑦 ( 𝑦 ∈ suc 𝑥 ∧ ¬ 𝑦 ∈ 𝐵 ) → ( 𝐵 ⊊ suc 𝑥 → ( 𝑥 ∈ ω → 𝐵 ≼ 𝑥 ) ) )
36 11 35 mpcom ⊢ ( 𝐵 ⊊ suc 𝑥 → ( 𝑥 ∈ ω → 𝐵 ≼ 𝑥 ) )
37 endomtr ⊢ ( ( suc 𝑥 ≈ 𝐵 ∧ 𝐵 ≼ 𝑥 ) → suc 𝑥 ≼ 𝑥 )
38 sssucid ⊢ 𝑥 ⊆ suc 𝑥
39 ssdomg ⊢ ( suc 𝑥 ∈ V → ( 𝑥 ⊆ suc 𝑥 → 𝑥 ≼ suc 𝑥 ) )
40 21 38 39 mp2 ⊢ 𝑥 ≼ suc 𝑥
41 sbth ⊢ ( ( suc 𝑥 ≼ 𝑥 ∧ 𝑥 ≼ suc 𝑥 ) → suc 𝑥 ≈ 𝑥 )
42 37 40 41 sylancl ⊢ ( ( suc 𝑥 ≈ 𝐵 ∧ 𝐵 ≼ 𝑥 ) → suc 𝑥 ≈ 𝑥 )
43 42 expcom ⊢ ( 𝐵 ≼ 𝑥 → ( suc 𝑥 ≈ 𝐵 → suc 𝑥 ≈ 𝑥 ) )
44 peano2b ⊢ ( 𝑥 ∈ ω ↔ suc 𝑥 ∈ ω )
45 nnord ⊢ ( suc 𝑥 ∈ ω → Ord suc 𝑥 )
46 44 45 sylbi ⊢ ( 𝑥 ∈ ω → Ord suc 𝑥 )
47 20 sucid ⊢ 𝑥 ∈ suc 𝑥
48 nordeq ⊢ ( ( Ord suc 𝑥 ∧ 𝑥 ∈ suc 𝑥 ) → suc 𝑥 ≠ 𝑥 )
49 46 47 48 sylancl ⊢ ( 𝑥 ∈ ω → suc 𝑥 ≠ 𝑥 )
50 nneneq ⊢ ( ( suc 𝑥 ∈ ω ∧ 𝑥 ∈ ω ) → ( suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥 ) )
51 44 50 sylanb ⊢ ( ( 𝑥 ∈ ω ∧ 𝑥 ∈ ω ) → ( suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥 ) )
52 51 anidms ⊢ ( 𝑥 ∈ ω → ( suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 = 𝑥 ) )
53 52 necon3bbid ⊢ ( 𝑥 ∈ ω → ( ¬ suc 𝑥 ≈ 𝑥 ↔ suc 𝑥 ≠ 𝑥 ) )
54 49 53 mpbird ⊢ ( 𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝑥 )
55 43 54 nsyli ⊢ ( 𝐵 ≼ 𝑥 → ( 𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝐵 ) )
56 36 55 syli ⊢ ( 𝐵 ⊊ suc 𝑥 → ( 𝑥 ∈ ω → ¬ suc 𝑥 ≈ 𝐵 ) )
57 56 com12 ⊢ ( 𝑥 ∈ ω → ( 𝐵 ⊊ suc 𝑥 → ¬ suc 𝑥 ≈ 𝐵 ) )
58 psseq2 ⊢ ( 𝐴 = suc 𝑥 → ( 𝐵 ⊊ 𝐴 ↔ 𝐵 ⊊ suc 𝑥 ) )
59 breq1 ⊢ ( 𝐴 = suc 𝑥 → ( 𝐴 ≈ 𝐵 ↔ suc 𝑥 ≈ 𝐵 ) )
60 59 notbid ⊢ ( 𝐴 = suc 𝑥 → ( ¬ 𝐴 ≈ 𝐵 ↔ ¬ suc 𝑥 ≈ 𝐵 ) )
61 58 60 imbi12d ⊢ ( 𝐴 = suc 𝑥 → ( ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵 ) ↔ ( 𝐵 ⊊ suc 𝑥 → ¬ suc 𝑥 ≈ 𝐵 ) ) )
62 57 61 syl5ibrcom ⊢ ( 𝑥 ∈ ω → ( 𝐴 = suc 𝑥 → ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵 ) ) )
63 62 rexlimiv ⊢ ( ∃ 𝑥 ∈ ω 𝐴 = suc 𝑥 → ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵 ) )
64 10 63 syl ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵 ) )
65 64 ex ⊢ ( 𝐴 ∈ ω → ( 𝐵 ⊊ 𝐴 → ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵 ) ) )
66 65 pm2.43d ⊢ ( 𝐴 ∈ ω → ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 ≈ 𝐵 ) )
67 66 imp ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ⊊ 𝐴 ) → ¬ 𝐴 ≈ 𝐵 )