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Theorem dif1enen 6882
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 6763 . . 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 4009 . . . . . . 7 (𝑛 = ∅ → (𝐴𝑛𝐴 ≈ ∅))
65adantl 277 . . . . . 6 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → (𝐴𝑛𝐴 ≈ ∅))
74, 6mpbid 147 . . . . 5 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ ∅)
8 en0 6797 . . . . 5 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
97, 8sylib 122 . . . 4 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴 = ∅)
10 dif1enen.c . . . . . 6 (𝜑𝐶𝐴)
11 n0i 3430 . . . . . 6 (𝐶𝐴 → ¬ 𝐴 = ∅)
1210, 11syl 14 . . . . 5 (𝜑 → ¬ 𝐴 = ∅)
1312ad2antrr 488 . . . 4 (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → ¬ 𝐴 = ∅)
149, 13pm2.21dd 620 . . 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 4009 . . . . . . . . . 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 6881 . . . . . . . 8 ((𝑚 ∈ ω ∧ 𝐴 ≈ suc 𝑚𝐶𝐴) → (𝐴 ∖ {𝐶}) ≈ 𝑚)
2315, 20, 21, 22syl3anc 1238 . . . . . . 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 6785 . . . . . . . . . 10 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐵𝐴)
27 entr 6786 . . . . . . . . . 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 6881 . . . . . . . . 9 ((𝑚 ∈ ω ∧ 𝐵 ≈ suc 𝑚𝐷𝐵) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
3215, 28, 30, 31syl3anc 1238 . . . . . . . 8 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
3332ensymd 6785 . . . . . . 7 ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ≈ (𝐵 ∖ {𝐷}))
34 entr 6786 . . . . . . 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 2594 . . . 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 4605 . . . 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 798 . 2 ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
423, 41rexlimddv 2599 1 (𝜑 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wcel 2148  wrex 2456  cdif 3128  c0 3424  {csn 3594   class class class wbr 4005  suc csuc 4367  ωcom 4591  cen 6740  Fincfn 6742
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-er 6537  df-en 6743  df-fin 6745
This theorem is referenced by:  fisseneq  6933
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