Theorem phplem3 3405

Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus one of its elements.

Hypotheses

Ref	Expression
phplem3.1	⊢ A ∈ V
phplem3.2	⊢ B ∈ V

Assertion

Ref	Expression
phplem3	⊢ ((A ∈ ω ∧ B ∈ A) → A ≈ (suc A ∖ {B}))

Proof of Theorem phplem3

Step	Hyp	Ref	Expression
1		difss 1596	. . . . . . . . . 10 ⊢ (A ∖ {B}) ⊆ A
2		ssrin 1661	. . . . . . . . . 10 ⊢ ((A ∖ {B}) ⊆ A → ((A ∖ {B}) ∩ {A}) ⊆ (A ∩ {A}))
3	1, 2	ax-mp 6	. . . . . . . . 9 ⊢ ((A ∖ {B}) ∩ {A}) ⊆ (A ∩ {A})
4		nnord 2381	. . . . . . . . . . 11 ⊢ (A ∈ ω → Ord A)
5		orddisj 2236	. . . . . . . . . . 11 ⊢ (Ord A → (A ∩ {A}) = ∅)
6	4, 5	syl 12	. . . . . . . . . 10 ⊢ (A ∈ ω → (A ∩ {A}) = ∅)
7	6	sseq2d 1528	. . . . . . . . 9 ⊢ (A ∈ ω → (((A ∖ {B}) ∩ {A}) ⊆ (A ∩ {A}) ↔ ((A ∖ {B}) ∩ {A}) ⊆ ∅))
8	3, 7	mpbii 168	. . . . . . . 8 ⊢ (A ∈ ω → ((A ∖ {B}) ∩ {A}) ⊆ ∅)
9		ss0 1727	. . . . . . . 8 ⊢ (((A ∖ {B}) ∩ {A}) ⊆ ∅ → ((A ∖ {B}) ∩ {A}) = ∅)
10	8, 9	syl 12	. . . . . . 7 ⊢ (A ∈ ω → ((A ∖ {B}) ∩ {A}) = ∅)
11		incom 1636	. . . . . . 7 ⊢ ({A} ∩ (A ∖ {B})) = ((A ∖ {B}) ∩ {A})
12	10, 11	syl5eq 1136	. . . . . 6 ⊢ (A ∈ ω → ({A} ∩ (A ∖ {B})) = ∅)
13		difdisj 1758	. . . . . 6 ⊢ ({B} ∩ (A ∖ {B})) = ∅
14	12, 13	jctil 240	. . . . 5 ⊢ (A ∈ ω → (({B} ∩ (A ∖ {B})) = ∅ ∧ ({A} ∩ (A ∖ {B})) = ∅))
15		phplem3.2	. . . . . . . 8 ⊢ B ∈ V
16		phplem3.1	. . . . . . . 8 ⊢ A ∈ V
17	15, 16	f1osn 2827	. . . . . . 7 ⊢ {⟨B, A⟩}:{B}–1-1-onto→{A}
18		snex 1859	. . . . . . . 8 ⊢ {B} ∈ V
19	18	f1oen 3301	. . . . . . 7 ⊢ ({⟨B, A⟩}:{B}–1-1-onto→{A} → {B} ≈ {A})
20	17, 19	ax-mp 6	. . . . . 6 ⊢ {B} ≈ {A}
21	16, 1	ssexi 1701	. . . . . . 7 ⊢ (A ∖ {B}) ∈ V
22	21	enref 3295	. . . . . 6 ⊢ (A ∖ {B}) ≈ (A ∖ {B})
23	20, 22	pm3.2i 234	. . . . 5 ⊢ ({B} ≈ {A} ∧ (A ∖ {B}) ≈ (A ∖ {B}))
24	14, 23	jctil 240	. . . 4 ⊢ (A ∈ ω → (({B} ≈ {A} ∧ (A ∖ {B}) ≈ (A ∖ {B})) ∧ (({B} ∩ (A ∖ {B})) = ∅ ∧ ({A} ∩ (A ∖ {B})) = ∅)))
25		unen 3338	. . . 4 ⊢ ((({B} ≈ {A} ∧ (A ∖ {B}) ≈ (A ∖ {B})) ∧ (({B} ∩ (A ∖ {B})) = ∅ ∧ ({A} ∩ (A ∖ {B})) = ∅)) → ({B} ∪ (A ∖ {B})) ≈ ({A} ∪ (A ∖ {B})))
26	24, 25	syl 12	. . 3 ⊢ (A ∈ ω → ({B} ∪ (A ∖ {B})) ≈ ({A} ∪ (A ∖ {B})))
27	26	adantr 306	. 2 ⊢ ((A ∈ ω ∧ B ∈ A) → ({B} ∪ (A ∖ {B})) ≈ ({A} ∪ (A ∖ {B})))
28		phplem1 3403	. . 3 ⊢ (B ∈ A → ({B} ∪ (A ∖ {B})) = A)
29	28	adantl 305	. 2 ⊢ ((A ∈ ω ∧ B ∈ A) → ({B} ∪ (A ∖ {B})) = A)
30		phplem2 3404	. 2 ⊢ ((A ∈ ω ∧ B ∈ A) → ({A} ∪ (A ∖ {B})) = (suc A ∖ {B}))
31	27, 29, 30	3brtr3d 2086	1 ⊢ ((A ∈ ω ∧ B ∈ A) → A ≈ (suc A ∖ {B}))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  {csn 1808  ⟨cop 1810   class class class wbr 2054  Ord word 2198  suc csuc 2201  ωcom 2372  –1-1-onto→wf1o 2421   ≈ cen 3271

This theorem is referenced by:  phplem4 3406

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-en 3274

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