Theorem dif1enen 6977

Description: Subtracting one element from each of two equinumerous finite sets. (Contributed by Jim Kingdon, 5-Jun-2022.)

Hypotheses

Ref	Expression
dif1enen.a	⊢ (𝜑 → 𝐴 ∈ Fin)
dif1enen.ab	⊢ (𝜑 → 𝐴 ≈ 𝐵)
dif1enen.c	⊢ (𝜑 → 𝐶 ∈ 𝐴)
dif1enen.d	⊢ (𝜑 → 𝐷 ∈ 𝐵)

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.

Step	Hyp	Ref	Expression
1		dif1enen.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
2		isfi 6852	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
3	1, 2	sylib 122	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
4		simplrr 536	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ 𝑛)
5		breq2 4048	. . . . . . 7 ⊢ (𝑛 = ∅ → (𝐴 ≈ 𝑛 ↔ 𝐴 ≈ ∅))
6	5	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → (𝐴 ≈ 𝑛 ↔ 𝐴 ≈ ∅))
7	4, 6	mpbid 147	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ ∅)
8		en0 6887	. . . . 5 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
9	7, 8	sylib 122	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝐴 = ∅)
10		dif1enen.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
11		n0i 3466	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → ¬ 𝐴 = ∅)
12	10, 11	syl 14	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 = ∅)
13	12	ad2antrr 488	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → ¬ 𝐴 = ∅)
14	9, 13	pm2.21dd 621	. . 3 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
15		simplr 528	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ∈ ω)
16		simprr 531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → 𝐴 ≈ 𝑛)
17	16	ad2antrr 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴 ≈ 𝑛)
18		breq2 4048	. . . . . . . . . 10 ⊢ (𝑛 = suc 𝑚 → (𝐴 ≈ 𝑛 ↔ 𝐴 ≈ suc 𝑚))
19	18	adantl 277	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐴 ≈ 𝑛 ↔ 𝐴 ≈ suc 𝑚))
20	17, 19	mpbid 147	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴 ≈ suc 𝑚)
21	10	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐶 ∈ 𝐴)
22		dif1en 6976	. . . . . . . 8 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ suc 𝑚 ∧ 𝐶 ∈ 𝐴) → (𝐴 ∖ {𝐶}) ≈ 𝑚)
23	15, 20, 21, 22	syl3anc 1250	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐴 ∖ {𝐶}) ≈ 𝑚)
24		dif1enen.ab	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≈ 𝐵)
25	24	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴 ≈ 𝐵)
26	25	ensymd 6875	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐵 ≈ 𝐴)
27		entr 6876	. . . . . . . . . 10 ⊢ ((𝐵 ≈ 𝐴 ∧ 𝐴 ≈ suc 𝑚) → 𝐵 ≈ suc 𝑚)
28	26, 20, 27	syl2anc 411	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐵 ≈ suc 𝑚)
29		dif1enen.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝐵)
30	29	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐷 ∈ 𝐵)
31		dif1en 6976	. . . . . . . . 9 ⊢ ((𝑚 ∈ ω ∧ 𝐵 ≈ suc 𝑚 ∧ 𝐷 ∈ 𝐵) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
32	15, 28, 30, 31	syl3anc 1250	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
33	32	ensymd 6875	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ≈ (𝐵 ∖ {𝐷}))
34		entr 6876	. . . . . . 7 ⊢ (((𝐴 ∖ {𝐶}) ≈ 𝑚 ∧ 𝑚 ≈ (𝐵 ∖ {𝐷})) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
35	23, 33, 34	syl2anc 411	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
36	35	ex 115	. . . . 5 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) → (𝑛 = suc 𝑚 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷})))
37	36	rexlimdva 2623	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (∃𝑚 ∈ ω 𝑛 = suc 𝑚 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷})))
38	37	imp 124	. . 3 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ ∃𝑚 ∈ ω 𝑛 = suc 𝑚) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
39		nn0suc 4652	. . . 4 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
40	39	ad2antrl 490	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
41	14, 38, 40	mpjaodan 800	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
42	3, 41	rexlimddv 2628	1 ⊢ (𝜑 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710   = wceq 1373   ∈ wcel 2176  ∃wrex 2485   ∖ cdif 3163  ∅c0 3460  {csn 3633   class class class wbr 4044  suc csuc 4412  ωcom 4638   ≈ cen 6825  Fincfn 6827

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6620  df-en 6828  df-fin 6830

This theorem is referenced by:  fisseneq  7031