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Theorem dif1enen 6676
 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 6558 . . 3 (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴𝑛)
31, 2sylib 121 . 2 (𝜑 → ∃𝑛 ∈ ω 𝐴𝑛)
4 simplrr 504 . . . . . 6 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴𝑛)
5 breq2 3871 . . . . . . 7 (𝑛 = ∅ → (𝐴𝑛𝐴 ≈ ∅))
65adantl 272 . . . . . 6 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → (𝐴𝑛𝐴 ≈ ∅))
74, 6mpbid 146 . . . . 5 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ ∅)
8 en0 6592 . . . . 5 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
97, 8sylib 121 . . . 4 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴 = ∅)
10 dif1enen.c . . . . . 6 (𝜑𝐶𝐴)
11 n0i 3307 . . . . . 6 (𝐶𝐴 → ¬ 𝐴 = ∅)
1210, 11syl 14 . . . . 5 (𝜑 → ¬ 𝐴 = ∅)
1312ad2antrr 473 . . . 4 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → ¬ 𝐴 = ∅)
149, 13pm2.21dd 588 . . 3 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
15 simplr 498 . . . . . . . 8 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ∈ ω)
16 simprr 500 . . . . . . . . . 10 ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → 𝐴𝑛)
1716ad2antrr 473 . . . . . . . . 9 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴𝑛)
18 breq2 3871 . . . . . . . . . 10 (𝑛 = suc 𝑚 → (𝐴𝑛𝐴 ≈ suc 𝑚))
1918adantl 272 . . . . . . . . 9 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐴𝑛𝐴 ≈ suc 𝑚))
2017, 19mpbid 146 . . . . . . . 8 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴 ≈ suc 𝑚)
2110ad3antrrr 477 . . . . . . . 8 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐶𝐴)
22 dif1en 6675 . . . . . . . 8 ((𝑚 ∈ ω ∧ 𝐴 ≈ suc 𝑚𝐶𝐴) → (𝐴 ∖ {𝐶}) ≈ 𝑚)
2315, 20, 21, 22syl3anc 1181 . . . . . . 7 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐴 ∖ {𝐶}) ≈ 𝑚)
24 dif1enen.ab . . . . . . . . . . . 12 (𝜑𝐴𝐵)
2524ad3antrrr 477 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴𝐵)
2625ensymd 6580 . . . . . . . . . 10 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐵𝐴)
27 entr 6581 . . . . . . . . . 10 ((𝐵𝐴𝐴 ≈ suc 𝑚) → 𝐵 ≈ suc 𝑚)
2826, 20, 27syl2anc 404 . . . . . . . . 9 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐵 ≈ suc 𝑚)
29 dif1enen.d . . . . . . . . . 10 (𝜑𝐷𝐵)
3029ad3antrrr 477 . . . . . . . . 9 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐷𝐵)
31 dif1en 6675 . . . . . . . . 9 ((𝑚 ∈ ω ∧ 𝐵 ≈ suc 𝑚𝐷𝐵) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
3215, 28, 30, 31syl3anc 1181 . . . . . . . 8 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
3332ensymd 6580 . . . . . . 7 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ≈ (𝐵 ∖ {𝐷}))
34 entr 6581 . . . . . . 7 (((𝐴 ∖ {𝐶}) ≈ 𝑚𝑚 ≈ (𝐵 ∖ {𝐷})) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
3523, 33, 34syl2anc 404 . . . . . 6 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
3635ex 114 . . . . 5 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) → (𝑛 = suc 𝑚 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷})))
3736rexlimdva 2502 . . . 4 ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (∃𝑚 ∈ ω 𝑛 = suc 𝑚 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷})))
3837imp 123 . . 3 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
39 nn0suc 4447 . . . 4 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4039ad2antrl 475 . . 3 ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
4114, 38, 40mpjaodan 750 . 2 ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
423, 41rexlimddv 2507 1 (𝜑 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 667   = wceq 1296   ∈ wcel 1445  ∃wrex 2371   ∖ cdif 3010  ∅c0 3302  {csn 3466   class class class wbr 3867  suc csuc 4216  ωcom 4433   ≈ cen 6535  Fincfn 6537 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-iinf 4431 This theorem depends on definitions:  df-bi 116  df-dc 784  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-if 3414  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-tr 3959  df-id 4144  df-iord 4217  df-on 4219  df-suc 4222  df-iom 4434  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-er 6332  df-en 6538  df-fin 6540 This theorem is referenced by:  fisseneq  6722
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