Theorem phplem1 4494

Description: Lemma for Pigeonhole Principle. If we join a natural number to itself minus an element, we end up with its successor minus the same element.

Assertion

Ref	Expression
phplem1	|- ((A e. om /\ B e. A) -> ({A} u. (A \ {B})) = (suc A \ {B}))

Proof of Theorem phplem1

Step	Hyp	Ref	Expression
1		nordeq 2962	. . . 4 |- ((Ord A /\ B e. A) -> A =/= B)
2		disjsn2 2438	. . . 4 |- (A =/= B -> ({A} i^i {B}) = (/))
3	1, 2	syl 10	. . 3 |- ((Ord A /\ B e. A) -> ({A} i^i {B}) = (/))
4		nnord 3135	. . 3 |- (A e. om -> Ord A)
5	3, 4	sylan 448	. 2 |- ((A e. om /\ B e. A) -> ({A} i^i {B}) = (/))
6		undif4 2321	. . 3 |- (({A} i^i {B}) = (/) -> ({A} u. (A \ {B})) = (({A} u. A) \ {B}))
7		df-suc 2949	. . . . 5 |- suc A = (A u. {A})
8		uncom 2172	. . . . 5 |- (A u. {A}) = ({A} u. A)
9	7, 8	eqtr 1492	. . . 4 |- suc A = ({A} u. A)
10	9	difeq1i 2151	. . 3 |- (suc A \ {B}) = (({A} u. A) \ {B})
11	6, 10	syl6eqr 1522	. 2 |- (({A} i^i {B}) = (/) -> ({A} u. (A \ {B})) = (suc A \ {B}))
12	5, 11	syl 10	1 |- ((A e. om /\ B e. A) -> ({A} u. (A \ {B})) = (suc A \ {B}))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956   =/= wne 1582   \ cdif 2040   u. cun 2041   i^i cin 2042  (/)c0 2276  {csn 2405  Ord word 2942  suc csuc 2945  omcom 3126

This theorem is referenced by:  phplem2 4495

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-suc 2949  df-om 3127

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