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Theorem dif1enen 6908
Description: Subtracting one element from each of two equinumerous finite sets. (Contributed by Jim Kingdon, 5-Jun-2022.)
Hypotheses
Ref Expression
dif1enen.a (𝜑𝐴 ∈ Fin)
dif1enen.ab (𝜑𝐴𝐵)
dif1enen.c (𝜑𝐶𝐴)
dif1enen.d (𝜑𝐷𝐵)
Assertion
Ref Expression
dif1enen (𝜑 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))

Proof of Theorem dif1enen
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dif1enen.a . . 3 (𝜑𝐴 ∈ Fin)
2 isfi 6787 . . 3 (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴𝑛)
31, 2sylib 122 . 2 (𝜑 → ∃𝑛 ∈ ω 𝐴𝑛)
4 simplrr 536 . . . . . 6 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴𝑛)
5 breq2 4022 . . . . . . 7 (𝑛 = ∅ → (𝐴𝑛𝐴 ≈ ∅))
65adantl 277 . . . . . 6 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → (𝐴𝑛𝐴 ≈ ∅))
74, 6mpbid 147 . . . . 5 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ ∅)
8 en0 6821 . . . . 5 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
97, 8sylib 122 . . . 4 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴 = ∅)
10 dif1enen.c . . . . . 6 (𝜑𝐶𝐴)
11 n0i 3443 . . . . . 6 (𝐶𝐴 → ¬ 𝐴 = ∅)
1210, 11syl 14 . . . . 5 (𝜑 → ¬ 𝐴 = ∅)
1312ad2antrr 488 . . . 4 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → ¬ 𝐴 = ∅)
149, 13pm2.21dd 621 . . 3 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
15 simplr 528 . . . . . . . 8 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ∈ ω)
16 simprr 531 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → 𝐴𝑛)
1716ad2antrr 488 . . . . . . . . 9 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴𝑛)
18 breq2 4022 . . . . . . . . . 10 (𝑛 = suc 𝑚 → (𝐴𝑛𝐴 ≈ suc 𝑚))
1918adantl 277 . . . . . . . . 9 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐴𝑛𝐴 ≈ suc 𝑚))
2017, 19mpbid 147 . . . . . . . 8 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴 ≈ suc 𝑚)
2110ad3antrrr 492 . . . . . . . 8 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐶𝐴)
22 dif1en 6907 . . . . . . . 8 ((𝑚 ∈ ω ∧ 𝐴 ≈ suc 𝑚𝐶𝐴) → (𝐴 ∖ {𝐶}) ≈ 𝑚)
2315, 20, 21, 22syl3anc 1249 . . . . . . 7 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐴 ∖ {𝐶}) ≈ 𝑚)
24 dif1enen.ab . . . . . . . . . . . 12 (𝜑𝐴𝐵)
2524ad3antrrr 492 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴𝐵)
2625ensymd 6809 . . . . . . . . . 10 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐵𝐴)
27 entr 6810 . . . . . . . . . 10 ((𝐵𝐴𝐴 ≈ suc 𝑚) → 𝐵 ≈ suc 𝑚)
2826, 20, 27syl2anc 411 . . . . . . . . 9 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐵 ≈ suc 𝑚)
29 dif1enen.d . . . . . . . . . 10 (𝜑𝐷𝐵)
3029ad3antrrr 492 . . . . . . . . 9 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐷𝐵)
31 dif1en 6907 . . . . . . . . 9 ((𝑚 ∈ ω ∧ 𝐵 ≈ suc 𝑚𝐷𝐵) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
3215, 28, 30, 31syl3anc 1249 . . . . . . . 8 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
3332ensymd 6809 . . . . . . 7 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ≈ (𝐵 ∖ {𝐷}))
34 entr 6810 . . . . . . 7 (((𝐴 ∖ {𝐶}) ≈ 𝑚𝑚 ≈ (𝐵 ∖ {𝐷})) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
3523, 33, 34syl2anc 411 . . . . . 6 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
3635ex 115 . . . . 5 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) → (𝑛 = suc 𝑚 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷})))
3736rexlimdva 2607 . . . 4 ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (∃𝑚 ∈ ω 𝑛 = suc 𝑚 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷})))
3837imp 124 . . 3 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
39 nn0suc 4621 . . . 4 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4039ad2antrl 490 . . 3 ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4114, 38, 40mpjaodan 799 . 2 ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
423, 41rexlimddv 2612 1 (𝜑 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709   = wceq 1364  wcel 2160  wrex 2469  cdif 3141  c0 3437  {csn 3607   class class class wbr 4018  suc csuc 4383  ωcom 4607  cen 6764  Fincfn 6766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-er 6559  df-en 6767  df-fin 6769
This theorem is referenced by:  fisseneq  6960
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