Theorem phplem1 4508

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 2967	. . . 4 |- ((Ord A /\ B e. A) -> A =/= B)
2		disjsn2 2442	. . . 4 |- (A =/= B -> ({A} i^i {B}) = (/))
3	1, 2	syl 10	. . 3 |- ((Ord A /\ B e. A) -> ({A} i^i {B}) = (/))
4		nnord 3140	. . 3 |- (A e. om -> Ord A)
5	3, 4	sylan 448	. 2 |- ((A e. om /\ B e. A) -> ({A} i^i {B}) = (/))
6		undif4 2325	. . 3 |- (({A} i^i {B}) = (/) -> ({A} u. (A \ {B})) = (({A} u. A) \ {B}))
7		df-suc 2954	. . . . 5 |- suc A = (A u. {A})
8		uncom 2176	. . . . 5 |- (A u. {A}) = ({A} u. A)
9	7, 8	eqtr 1495	. . . 4 |- suc A = ({A} u. A)
10	9	difeq1i 2155	. . 3 |- (suc A \ {B}) = (({A} u. A) \ {B})
11	6, 10	syl6eqr 1525	. 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 956   e. wcel 958   =/= wne 1585   \ cdif 2044   u. cun 2045   i^i cin 2046  (/)c0 2280  {csn 2409  Ord word 2947  suc csuc 2950  omcom 3131

This theorem is referenced by:  phplem2 4509

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-om 3132

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