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Theorem dif1en 8138
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 7034 . . . . 5 (𝑀 ∈ ω → suc 𝑀 ∈ ω)
2 breq2 4622 . . . . . . 7 (𝑥 = suc 𝑀 → (𝐴𝑥𝐴 ≈ suc 𝑀))
32rspcev 3300 . . . . . 6 ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → ∃𝑥 ∈ ω 𝐴𝑥)
4 isfi 7924 . . . . . 6 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
53, 4sylibr 224 . . . . 5 ((suc 𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
61, 5sylan 488 . . . 4 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) → 𝐴 ∈ Fin)
7 diffi 8137 . . . . 5 (𝐴 ∈ Fin → (𝐴 ∖ {𝑋}) ∈ Fin)
8 isfi 7924 . . . . 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 1079 . 2 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → ∃𝑥 ∈ ω (𝐴 ∖ {𝑋}) ≈ 𝑥)
12 vex 3194 . . . . . . . 8 𝑥 ∈ V
13 en2sn 7982 . . . . . . . 8 ((𝑋𝐴𝑥 ∈ V) → {𝑋} ≈ {𝑥})
1412, 13mpan2 706 . . . . . . 7 (𝑋𝐴 → {𝑋} ≈ {𝑥})
15 nnord 7021 . . . . . . . 8 (𝑥 ∈ ω → Ord 𝑥)
16 orddisj 5724 . . . . . . . 8 (Ord 𝑥 → (𝑥 ∩ {𝑥}) = ∅)
1715, 16syl 17 . . . . . . 7 (𝑥 ∈ ω → (𝑥 ∩ {𝑥}) = ∅)
18 incom 3788 . . . . . . . . . 10 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ({𝑋} ∩ (𝐴 ∖ {𝑋}))
19 disjdif 4017 . . . . . . . . . 10 ({𝑋} ∩ (𝐴 ∖ {𝑋})) = ∅
2018, 19eqtri 2648 . . . . . . . . 9 ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅
21 unen 7985 . . . . . . . . . 10 ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ {𝑋} ≈ {𝑥}) ∧ (((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅ ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2221an4s 868 . . . . . . . . 9 ((((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ((𝐴 ∖ {𝑋}) ∩ {𝑋}) = ∅) ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2320, 22mpanl2 716 . . . . . . . 8 (((𝐴 ∖ {𝑋}) ≈ 𝑥 ∧ ({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅)) → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}))
2423expcom 451 . . . . . . 7 (({𝑋} ≈ {𝑥} ∧ (𝑥 ∩ {𝑥}) = ∅) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
2514, 17, 24syl2an 494 . . . . . 6 ((𝑋𝐴𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
26253ad2antl3 1223 . . . . 5 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → ((𝐴 ∖ {𝑋}) ≈ 𝑥 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥})))
27 difsnid 4315 . . . . . . . . 9 (𝑋𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
28 df-suc 5691 . . . . . . . . . . 11 suc 𝑥 = (𝑥 ∪ {𝑥})
2928eqcomi 2635 . . . . . . . . . 10 (𝑥 ∪ {𝑥}) = suc 𝑥
3029a1i 11 . . . . . . . . 9 (𝑋𝐴 → (𝑥 ∪ {𝑥}) = suc 𝑥)
3127, 30breq12d 4631 . . . . . . . 8 (𝑋𝐴 → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
32313ad2ant3 1082 . . . . . . 7 ((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
3332adantr 481 . . . . . 6 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀𝑋𝐴) ∧ 𝑥 ∈ ω) → (((𝐴 ∖ {𝑋}) ∪ {𝑋}) ≈ (𝑥 ∪ {𝑥}) ↔ 𝐴 ≈ suc 𝑥))
34 ensym 7950 . . . . . . . . . . 11 (𝐴 ≈ suc 𝑀 → suc 𝑀𝐴)
35 entr 7953 . . . . . . . . . . . . 13 ((suc 𝑀𝐴𝐴 ≈ suc 𝑥) → suc 𝑀 ≈ suc 𝑥)
36 peano2 7034 . . . . . . . . . . . . . 14 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
37 nneneq 8088 . . . . . . . . . . . . . 14 ((suc 𝑀 ∈ ω ∧ suc 𝑥 ∈ ω) → (suc 𝑀 ≈ suc 𝑥 ↔ suc 𝑀 = suc 𝑥))
3836, 37sylan2 491 . . . . . . . . . . . . 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 452 . . . . . . . . . . 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 488 . . . . . . . . 9 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑀 → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥)))
4342imp 445 . . . . . . . 8 (((𝑀 ∈ ω ∧ 𝑥 ∈ ω) ∧ 𝐴 ≈ suc 𝑀) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
4443an32s 845 . . . . . . 7 (((𝑀 ∈ ω ∧ 𝐴 ≈ suc 𝑀) ∧ 𝑥 ∈ ω) → (𝐴 ≈ suc 𝑥 → suc 𝑀 = suc 𝑥))
45443adantl3 1217 . . . . . 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 7036 . . . . . . 7 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
4847biimpd 219 . . . . . 6 ((𝑀 ∈ ω ∧ 𝑥 ∈ ω) → (suc 𝑀 = suc 𝑥𝑀 = 𝑥))
49483ad2antl1 1221 . . . . 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 4622 . . . . 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 3029 . 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 384  w3a 1036   = wceq 1480  wcel 1992  wrex 2913  Vcvv 3191  cdif 3557  cun 3558  cin 3559  c0 3896  {csn 4153   class class class wbr 4618  Ord word 5684  suc csuc 5687  ωcom 7013  cen 7897  Fincfn 7900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-om 7014  df-1o 7506  df-er 7688  df-en 7901  df-fin 7904
This theorem is referenced by:  enp1i  8140  findcard  8144  findcard2  8145  en2eleq  8776  en2other2  8777  mreexexlem4d  16223  f1otrspeq  17783  pmtrf  17791  pmtrmvd  17792  pmtrfinv  17797
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