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Theorem dif1en 8234
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 7128 . . . . 5 (𝑀 ∈ ω → suc 𝑀 ∈ ω)
2 breq2 4689 . . . . . . 7 (𝑥 = suc 𝑀 → (𝐴𝑥𝐴 ≈ suc 𝑀))
32rspcev 3340 . . . . . 6 ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω 𝐴𝑥)
4 isfi 8021 . . . . . 6 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
53, 4sylibr 224 . . . . 5 ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
61, 5sylan 487 . . . 4 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
7 diffi 8233 . . . . 5 (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
8 isfi 8021 . . . . 5 ((𝐴 ∖ {𝑋}) ∈ Fin ↔ ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
97, 8sylib 208 . . . 4 (𝐴 ∈ Fin → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
106, 9syl 17 . . 3 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
11103adant3 1101 . 2 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
12 vex 3234 . . . . . . . 8 𝑥 ∈ V
13 en2sn 8078 . . . . . . . 8 ((𝑋𝐴𝑥 ∈ V) → {𝑋} ≈ {𝑥})
1412, 13mpan2 707 . . . . . . 7 (𝑋𝐴 → {𝑋} ≈ {𝑥})
15 nnord 7115 . . . . . . . 8 (𝑥 ∈ ω → Ord 𝑥)
16 orddisj 5800 . . . . . . . 8 (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
1715, 16syl 17 . . . . . . 7 (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
18 incom 3838 . . . . . . . . . 10 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ({𝑋} ∩ (𝐴 ∖ {𝑋}))
19 disjdif 4073 . . . . . . . . . 10 ({𝑋} ∩ (𝐴 ∖ {𝑋})) = ∅
2018, 19eqtri 2673 . . . . . . . . 9 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
21 unen 8081 . . . . . . . . . 10 ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ {𝑋} ≈ {𝑥}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2221an4s 886 . . . . . . . . 9 ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅) ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2320, 22mpanl2 717 . . . . . . . 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 1245 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
27 difsnid 4373 . . . . . . . . 9 (𝑋𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
28 df-suc 5767 . . . . . . . . . . 11 suc 𝑥 = (𝑥 ∪ {𝑥})
2928eqcomi 2660 . . . . . . . . . 10 (𝑥 ∪ {𝑥}) = suc 𝑥
3029a1i 11 . . . . . . . . 9 (𝑋𝐴 → (𝑥 ∪ {𝑥}) = suc 𝑥)
3127, 30breq12d 4698 . . . . . . . 8 (𝑋𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
32313ad2ant3 1104 . . . . . . 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 8046 . . . . . . . . . . 11 (𝐴 ≈ suc 𝑀 → suc 𝑀𝐴)
35 entr 8049 . . . . . . . . . . . . 13 ((suc 𝑀𝐴𝐴 ≈ suc 𝑥) → suc 𝑀 ≈ suc 𝑥)
36 peano2 7128 . . . . . . . . . . . . . 14 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
37 nneneq 8184 . . . . . . . . . . . . . 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 234 . . . . . . . . . . . 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 34 . . . . . . . . . 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 863 . . . . . . 7 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
45443adantl3 1239 . . . . . 6 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
4633, 45sylbid 230 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) → suc 𝑀 = suc 𝑥))
47 peano4 7130 . . . . . . 7 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
4847biimpd 219 . . . . . 6 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
49483ad2antl1 1243 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
5026, 46, 493syld 60 . . . 4 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥𝑀 = 𝑥))
51 breq2 4689 . . . . 5 (𝑀 = 𝑥 → ((𝐴 ∖ {𝑋}) ≈ 𝑀 ↔ (𝐴 ∖ {𝑋}) ≈ 𝑥))
5251biimprcd 240 . . . 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 3060 . 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  Vcvv 3231  cdif 3604  cun 3605  cin 3606  c0 3948  {csn 4210   class class class wbr 4685  Ord word 5760  suc csuc 5763  ωcom 7107  cen 7994  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-fin 8001
This theorem is referenced by:  enp1i  8236  findcard  8240  findcard2  8241  en2eleq  8869  en2other2  8870  mreexexlem4d  16354  f1otrspeq  17913  pmtrf  17921  pmtrmvd  17922  pmtrfinv  17927
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