Theorem dif1enen 7112

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 6977	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
3	1, 2	sylib 122	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
4		simplrr 538	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ 𝑛)
5		breq2 4097	. . . . . . 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 7012	. . . . 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 3502	. . . . . 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 625	. . 3 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
15		simplr 529	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ∈ ω)
16		simprr 533	. . . . . . . . . 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 4097	. . . . . . . . . 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 7111	. . . . . . . 8 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ suc 𝑚 ∧ 𝐶 ∈ 𝐴) → (𝐴 ∖ {𝐶}) ≈ 𝑚)
23	15, 20, 21, 22	syl3anc 1274	. . . . . . 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 7000	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐵 ≈ 𝐴)
27		entr 7001	. . . . . . . . . 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 7111	. . . . . . . . 9 ⊢ ((𝑚 ∈ ω ∧ 𝐵 ≈ suc 𝑚 ∧ 𝐷 ∈ 𝐵) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
32	15, 28, 30, 31	syl3anc 1274	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
33	32	ensymd 7000	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ≈ (𝐵 ∖ {𝐷}))
34		entr 7001	. . . . . . 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 2651	. . . 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 4708	. . . 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 806	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
42	3, 41	rexlimddv 2656	1 ⊢ (𝜑 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716   = wceq 1398   ∈ wcel 2202  ∃wrex 2512   ∖ cdif 3198  ∅c0 3496  {csn 3673   class class class wbr 4093  suc csuc 4468  ωcom 4694   ≈ cen 6950  Fincfn 6952

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-er 6745  df-en 6953  df-fin 6955

This theorem is referenced by:  fisseneq  7170