Theorem phplem3 4510

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

Hypotheses

Ref	Expression
phplem2.1	|- A e. V
phplem2.2	|- B e. V

Assertion

Ref	Expression
phplem3	|- ((A e. om /\ B e. suc A) -> A ~~ (suc A \ {B}))

Proof of Theorem phplem3

Step	Hyp	Ref	Expression
1		phplem2.1	. . . 4 |- A e. V
2		phplem2.2	. . . 4 |- B e. V
3	1, 2	phplem2 4509	. . 3 |- ((A e. om /\ B e. A) -> A ~~ (suc A \ {B}))
4		nnord 3140	. . . . . 6 |- (A e. om -> Ord A)
5		orddif 3075	. . . . . 6 |- (Ord A -> A = (suc A \ {A}))
6	4, 5	syl 10	. . . . 5 |- (A e. om -> A = (suc A \ {A}))
7		sneq 2417	. . . . . . 7 |- (A = B -> {A} = {B})
8	7	difeq2d 2159	. . . . . 6 |- (A = B -> (suc A \ {A}) = (suc A \ {B}))
9	8	eqcoms 1478	. . . . 5 |- (B = A -> (suc A \ {A}) = (suc A \ {B}))
10	6, 9	sylan9eq 1527	. . . 4 |- ((A e. om /\ B = A) -> A = (suc A \ {B}))
11	1	enref 4391	. . . 4 |- A ~~ A
12	10, 11	syl5breq 2650	. . 3 |- ((A e. om /\ B = A) -> A ~~ (suc A \ {B}))
13	3, 12	jaodan 426	. 2 |- ((A e. om /\ (B e. A \/ B = A)) -> A ~~ (suc A \ {B}))
14		elsuci 3035	. 2 |- (B e. suc A -> (B e. A \/ B = A))
15	13, 14	sylan2 451	1 |- ((A e. om /\ B e. suc A) -> A ~~ (suc A \ {B}))

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   \ cdif 2044  {csn 2409   class class class wbr 2619  Ord word 2947  suc csuc 2950  omcom 3131   ~~ cen 4364

This theorem is referenced by:  phplem4 4511  php 4513

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-9 965  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-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

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-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-en 4368

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