Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dif1en Structured version   Visualization version   GIF version

Theorem dif1en 8078
 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 6978 . . . . 5 (𝑀 ∈ ω → suc 𝑀 ∈ ω)
2 breq2 4587 . . . . . . 7 (𝑥 = suc 𝑀 → (𝐴𝑥𝐴 ≈ suc 𝑀))
32rspcev 3282 . . . . . 6 ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω 𝐴𝑥)
4 isfi 7865 . . . . . 6 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
53, 4sylibr 223 . . . . 5 ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
61, 5sylan 487 . . . 4 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
7 diffi 8077 . . . . 5 (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
8 isfi 7865 . . . . 5 ((𝐴 ∖ {𝑋}) ∈ Fin ↔ ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
97, 8sylib 207 . . . 4 (𝐴 ∈ Fin → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
106, 9syl 17 . . 3 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
11103adant3 1074 . 2 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
12 vex 3176 . . . . . . . 8 𝑥 ∈ V
13 en2sn 7922 . . . . . . . 8 ((𝑋𝐴𝑥 ∈ V) → {𝑋} ≈ {𝑥})
1412, 13mpan2 703 . . . . . . 7 (𝑋𝐴 → {𝑋} ≈ {𝑥})
15 nnord 6965 . . . . . . . 8 (𝑥 ∈ ω → Ord 𝑥)
16 orddisj 5679 . . . . . . . 8 (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
1715, 16syl 17 . . . . . . 7 (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
18 incom 3767 . . . . . . . . . 10 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ({𝑋} ∩ (𝐴 ∖ {𝑋}))
19 disjdif 3992 . . . . . . . . . 10 ({𝑋} ∩ (𝐴 ∖ {𝑋})) = ∅
2018, 19eqtri 2632 . . . . . . . . 9 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
21 unen 7925 . . . . . . . . . 10 ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ {𝑋} ≈ {𝑥}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2221an4s 865 . . . . . . . . 9 ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅) ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2320, 22mpanl2 713 . . . . . . . 8 (((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2423expcom 450 . . . . . . 7 (({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
2514, 17, 24syl2an 493 . . . . . 6 ((𝑋𝐴𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
26253ad2antl3 1218 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
27 difsnid 4282 . . . . . . . . 9 (𝑋𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
28 df-suc 5646 . . . . . . . . . . 11 suc 𝑥 = (𝑥 ∪ {𝑥})
2928eqcomi 2619 . . . . . . . . . 10 (𝑥 ∪ {𝑥}) = suc 𝑥
3029a1i 11 . . . . . . . . 9 (𝑋𝐴 → (𝑥 ∪ {𝑥}) = suc 𝑥)
3127, 30breq12d 4596 . . . . . . . 8 (𝑋𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
32313ad2ant3 1077 . . . . . . 7 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
3332adantr 480 . . . . . 6 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
34 ensym 7891 . . . . . . . . . . 11 (𝐴 ≈ suc 𝑀 → suc 𝑀𝐴)
35 entr 7894 . . . . . . . . . . . . 13 ((suc 𝑀𝐴𝐴 ≈ suc 𝑥) → suc 𝑀 ≈ suc 𝑥)
36 peano2 6978 . . . . . . . . . . . . . 14 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
37 nneneq 8028 . . . . . . . . . . . . . 14 ((suc 𝑀 ∈ ω ∧ suc 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
3836, 37sylan2 490 . . . . . . . . . . . . 13 ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
3935, 38syl5ib 233 . . . . . . . . . . . 12 ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → ((suc 𝑀𝐴𝐴 ≈ suc 𝑥) → suc 𝑀 = suc 𝑥))
4039expd 451 . . . . . . . . . . 11 ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀𝐴 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
4134, 40syl5 33 . . . . . . . . . 10 ((suc 𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
421, 41sylan 487 . . . . . . . . 9 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
4342imp 444 . . . . . . . 8 (((𝑀 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ≈ suc 𝑀) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
4443an32s 842 . . . . . . 7 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
45443adantl3 1212 . . . . . 6 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
4633, 45sylbid 229 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) → suc 𝑀 = suc 𝑥))
47 peano4 6980 . . . . . . 7 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
4847biimpd 218 . . . . . 6 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
49483ad2antl1 1216 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
5026, 46, 493syld 58 . . . 4 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥𝑀 = 𝑥))
51 breq2 4587 . . . . 5 (𝑀 = 𝑥 → ((𝐴 ∖ {𝑋}) ≈ 𝑀 ↔ (𝐴 ∖ {𝑋}) ≈ 𝑥))
5251biimprcd 239 . . . 4 ((𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝑀 = 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
5350, 52sylcom 30 . . 3 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
5453rexlimdva 3013 . 2 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥 → (𝐴 ∖ {𝑋}) ≈ 𝑀))
5511, 54mpd 15 1 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (𝐴 ∖ {𝑋}) ≈ 𝑀)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  {csn 4125   class class class wbr 4583  Ord word 5639  suc csuc 5642  ωcom 6957   ≈ cen 7838  Fincfn 7841 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-er 7629  df-en 7842  df-fin 7845 This theorem is referenced by:  enp1i  8080  findcard  8084  findcard2  8085  en2eleq  8714  en2other2  8715  mreexexlem4d  16130  f1otrspeq  17690  pmtrf  17698  pmtrmvd  17699  pmtrfinv  17704
 Copyright terms: Public domain W3C validator