Theorem phplem2 4509

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

Hypotheses

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

Assertion

Ref	Expression
phplem2	|- ((A e. om /\ B e. A) -> A ~~ (suc A \ {B}))

Proof of Theorem phplem2

Step	Hyp	Ref	Expression
1		difss 2167	. . . . . . . . . 10 |- (A \ {B}) (_ A
2		ssrin 2234	. . . . . . . . . 10 |- ((A \ {B}) (_ A -> ((A \ {B}) i^i {A}) (_ (A i^i {A}))
3	1, 2	ax-mp 7	. . . . . . . . 9 |- ((A \ {B}) i^i {A}) (_ (A i^i {A})
4		nnord 3140	. . . . . . . . . . 11 |- (A e. om -> Ord A)
5		orddisj 2985	. . . . . . . . . . 11 |- (Ord A -> (A i^i {A}) = (/))
6	4, 5	syl 10	. . . . . . . . . 10 |- (A e. om -> (A i^i {A}) = (/))
7	6	sseq2d 2089	. . . . . . . . 9 |- (A e. om -> (((A \ {B}) i^i {A}) (_ (A i^i {A}) <-> ((A \ {B}) i^i {A}) (_ (/)))
8	3, 7	mpbii 193	. . . . . . . 8 |- (A e. om -> ((A \ {B}) i^i {A}) (_ (/))
9		ss0 2303	. . . . . . . 8 |- (((A \ {B}) i^i {A}) (_ (/) -> ((A \ {B}) i^i {A}) = (/))
10	8, 9	syl 10	. . . . . . 7 |- (A e. om -> ((A \ {B}) i^i {A}) = (/))
11		incom 2208	. . . . . . 7 |- ({A} i^i (A \ {B})) = ((A \ {B}) i^i {A})
12	10, 11	syl5eq 1519	. . . . . 6 |- (A e. om -> ({A} i^i (A \ {B})) = (/))
13		difdisj 2337	. . . . . 6 |- ({B} i^i (A \ {B})) = (/)
14	12, 13	jctil 292	. . . . 5 |- (A e. om -> (({B} i^i (A \ {B})) = (/) /\ ({A} i^i (A \ {B})) = (/)))
15		phplem2.2	. . . . . . . 8 |- B e. V
16		phplem2.1	. . . . . . . 8 |- A e. V
17	15, 16	f1osn 3719	. . . . . . 7 |- {<.B, A>.}:{B}-1-1-onto->{A}
18		snex 2750	. . . . . . . 8 |- {B} e. V
19	18	f1oen 4398	. . . . . . 7 |- ({<.B, A>.}:{B}-1-1-onto->{A} -> {B} ~~ {A})
20	17, 19	ax-mp 7	. . . . . 6 |- {B} ~~ {A}
21	16, 1	ssexi 2720	. . . . . . 7 |- (A \ {B}) e. V
22	21	enref 4391	. . . . . 6 |- (A \ {B}) ~~ (A \ {B})
23	20, 22	pm3.2i 285	. . . . 5 |- ({B} ~~ {A} /\ (A \ {B}) ~~ (A \ {B}))
24	14, 23	jctil 292	. . . 4 |- (A e. om -> (({B} ~~ {A} /\ (A \ {B}) ~~ (A \ {B})) /\ (({B} i^i (A \ {B})) = (/) /\ ({A} i^i (A \ {B})) = (/))))
25		unen 4434	. . . 4 |- ((({B} ~~ {A} /\ (A \ {B}) ~~ (A \ {B})) /\ (({B} i^i (A \ {B})) = (/) /\ ({A} i^i (A \ {B})) = (/))) -> ({B} u. (A \ {B})) ~~ ({A} u. (A \ {B})))
26	24, 25	syl 10	. . 3 |- (A e. om -> ({B} u. (A \ {B})) ~~ ({A} u. (A \ {B})))
27	26	adantr 389	. 2 |- ((A e. om /\ B e. A) -> ({B} u. (A \ {B})) ~~ ({A} u. (A \ {B})))
28		difsnid 2467	. . . 4 |- (B e. A -> ((A \ {B}) u. {B}) = A)
29		uncom 2176	. . . 4 |- ({B} u. (A \ {B})) = ((A \ {B}) u. {B})
30	28, 29	syl5eq 1519	. . 3 |- (B e. A -> ({B} u. (A \ {B})) = A)
31	30	adantl 388	. 2 |- ((A e. om /\ B e. A) -> ({B} u. (A \ {B})) = A)
32		phplem1 4508	. 2 |- ((A e. om /\ B e. A) -> ({A} u. (A \ {B})) = (suc A \ {B}))
33	27, 31, 32	3brtr3d 2644	1 |- ((A e. om /\ B e. A) -> A ~~ (suc A \ {B}))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   \ cdif 2044   u. cun 2045   i^i cin 2046   (_ wss 2047  (/)c0 2280  {csn 2409  <.cop 2411   class class class wbr 2619  Ord word 2947  suc csuc 2950  omcom 3131  -1-1-onto->wf1o 3181   ~~ cen 4364

This theorem is referenced by:  phplem3 4510

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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