Theorem phplem4 3406

Description: Lemma for Pigeonhole Principle. A natural number is equinumerous to its successor minus any element of the successor.

Hypotheses

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

Assertion

Ref	Expression
phplem4	⊢ ((A ∈ ω ∧ B ∈ suc A) → A ≈ (suc A ∖ {B}))

Proof of Theorem phplem4

Step	Hyp	Ref	Expression
1		andi 456	. . 3 ⊢ ((A ∈ ω ∧ (B ∈ A ∨ B = A)) ↔ ((A ∈ ω ∧ B ∈ A) ∨ (A ∈ ω ∧ B = A)))
2		phplem3.1	. . . . 5 ⊢ A ∈ V
3		phplem3.2	. . . . 5 ⊢ B ∈ V
4	2, 3	phplem3 3405	. . . 4 ⊢ ((A ∈ ω ∧ B ∈ A) → A ≈ (suc A ∖ {B}))
5		nnord 2381	. . . . . . 7 ⊢ (A ∈ ω → Ord A)
6		orddif 2326	. . . . . . 7 ⊢ (Ord A → A = (suc A ∖ {A}))
7	5, 6	syl 12	. . . . . 6 ⊢ (A ∈ ω → A = (suc A ∖ {A}))
8		sneq 1816	. . . . . . . 8 ⊢ (A = B → {A} = {B})
9	8	difeq2d 1588	. . . . . . 7 ⊢ (A = B → (suc A ∖ {A}) = (suc A ∖ {B}))
10	9	cleqcoms 1104	. . . . . 6 ⊢ (B = A → (suc A ∖ {A}) = (suc A ∖ {B}))
11	7, 10	sylan9eq 1144	. . . . 5 ⊢ ((A ∈ ω ∧ B = A) → A = (suc A ∖ {B}))
12	2	enref 3295	. . . . 5 ⊢ A ≈ A
13	11, 12	syl5breq 2091	. . . 4 ⊢ ((A ∈ ω ∧ B = A) → A ≈ (suc A ∖ {B}))
14	4, 13	jaoi 275	. . 3 ⊢ (((A ∈ ω ∧ B ∈ A) ∨ (A ∈ ω ∧ B = A)) → A ≈ (suc A ∖ {B}))
15	1, 14	sylbi 174	. 2 ⊢ ((A ∈ ω ∧ (B ∈ A ∨ B = A)) → A ≈ (suc A ∖ {B}))
16		elsuci 2289	. 2 ⊢ (B ∈ suc A → (B ∈ A ∨ B = A))
17	15, 16	sylan2 346	1 ⊢ ((A ∈ ω ∧ B ∈ suc A) → A ≈ (suc A ∖ {B}))

Syntax hints:   → wi 2   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ∖ cdif 1484  {csn 1808   class class class wbr 2054  Ord word 2198  suc csuc 2201  ωcom 2372   ≈ cen 3271

This theorem is referenced by:  phplem5 3407  php 3409

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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