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Theorem dif1en 8400
Description: If a set 𝐴 is equinumerous to the successor of a natural number 𝑀, then 𝐴 with an element removed is equinumerous to 𝑀. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Stefan O'Rear, 16-Aug-2015.)
Assertion
Ref Expression
dif1en ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)

Proof of Theorem dif1en
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 peano2 7284 . . . . 5 (𝑀 ∈ ω → suc 𝑀 ∈ ω)
2 breq2 4813 . . . . . . 7 (𝑥 = suc 𝑀 → (𝐴𝑥𝐴 ≈ suc 𝑀))
32rspcev 3461 . . . . . 6 ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω 𝐴𝑥)
4 isfi 8184 . . . . . 6 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
53, 4sylibr 225 . . . . 5 ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
61, 5sylan 575 . . . 4 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
7 diffi 8399 . . . . 5 (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
8 isfi 8184 . . . . 5 ((𝐴 ∖ {𝑋}) ∈ Fin ↔ ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
97, 8sylib 209 . . . 4 (𝐴 ∈ Fin → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
106, 9syl 17 . . 3 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
11103adant3 1162 . 2 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
12 en2sn 8244 . . . . . . . 8 ((𝑋𝐴𝑥 ∈ V) → {𝑋} ≈ {𝑥})
1312elvd 3355 . . . . . . 7 (𝑋𝐴 → {𝑋} ≈ {𝑥})
14 nnord 7271 . . . . . . . 8 (𝑥 ∈ ω → Ord 𝑥)
15 orddisj 5946 . . . . . . . 8 (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
1614, 15syl 17 . . . . . . 7 (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
17 incom 3967 . . . . . . . . . 10 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ({𝑋} ∩ (𝐴 ∖ {𝑋}))
18 disjdif 4200 . . . . . . . . . 10 ({𝑋} ∩ (𝐴 ∖ {𝑋})) = ∅
1917, 18eqtri 2787 . . . . . . . . 9 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
20 unen 8247 . . . . . . . . . 10 ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ {𝑋} ≈ {𝑥}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2120an4s 650 . . . . . . . . 9 ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅) ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2219, 21mpanl2 692 . . . . . . . 8 (((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2322expcom 402 . . . . . . 7 (({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
2413, 16, 23syl2an 589 . . . . . 6 ((𝑋𝐴𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
25243ad2antl3 1238 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
26 difsnid 4495 . . . . . . . . 9 (𝑋𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
27 df-suc 5914 . . . . . . . . . . 11 suc 𝑥 = (𝑥 ∪ {𝑥})
2827eqcomi 2774 . . . . . . . . . 10 (𝑥 ∪ {𝑥}) = suc 𝑥
2928a1i 11 . . . . . . . . 9 (𝑋𝐴 → (𝑥 ∪ {𝑥}) = suc 𝑥)
3026, 29breq12d 4822 . . . . . . . 8 (𝑋𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
31303ad2ant3 1165 . . . . . . 7 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
3231adantr 472 . . . . . 6 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
33 ensym 8209 . . . . . . . . . . 11 (𝐴 ≈ suc 𝑀 → suc 𝑀𝐴)
34 entr 8212 . . . . . . . . . . . . 13 ((suc 𝑀𝐴𝐴 ≈ suc 𝑥) → suc 𝑀 ≈ suc 𝑥)
35 peano2 7284 . . . . . . . . . . . . . 14 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
36 nneneq 8350 . . . . . . . . . . . . . 14 ((suc 𝑀 ∈ ω ∧ suc 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
3735, 36sylan2 586 . . . . . . . . . . . . 13 ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
3834, 37syl5ib 235 . . . . . . . . . . . 12 ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → ((suc 𝑀𝐴𝐴 ≈ suc 𝑥) → suc 𝑀 = suc 𝑥))
3938expd 404 . . . . . . . . . . 11 ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀𝐴 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
4033, 39syl5 34 . . . . . . . . . 10 ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
411, 40sylan 575 . . . . . . . . 9 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
4241imp 395 . . . . . . . 8 (((𝑀 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ≈ suc 𝑀) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
4342an32s 642 . . . . . . 7 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
44433adantl3 1209 . . . . . 6 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
4532, 44sylbid 231 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) → suc 𝑀 = suc 𝑥))
46 peano4 7286 . . . . . . 7 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
4746biimpd 220 . . . . . 6 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
48473ad2antl1 1236 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
4925, 45, 483syld 60 . . . 4 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥𝑀 = 𝑥))
50 breq2 4813 . . . . 5 (𝑀 = 𝑥 → ((𝐴 ∖ {𝑋}) ≈ 𝑀 ↔ (𝐴 ∖ {𝑋}) ≈ 𝑥))
5150biimprcd 241 . . . 4 ((𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝑀 = 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
5249, 51sylcom 30 . . 3 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
5352rexlimdva 3178 . 2 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
5411, 53mpd 15 1 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wrex 3056  Vcvv 3350  cdif 3729  cun 3730  cin 3731  c0 4079  {csn 4334   class class class wbr 4809  Ord word 5907  suc csuc 5910  ωcom 7263  cen 8157  Fincfn 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-1o 7764  df-er 7947  df-en 8161  df-fin 8164
This theorem is referenced by:  enp1i  8402  findcard  8406  findcard2  8407  en2eleq  9082  en2other2  9083  mreexexlem4d  16573  f1otrspeq  18130  pmtrf  18138  pmtrmvd  18139  pmtrfinv  18144
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