Theorem dif1enen 6879

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 6760	. . 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 4007	. . . . . . 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 6794	. . . . 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 3428	. . . . . 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 620	. . 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 4007	. . . . . . . . . 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 6878	. . . . . . . 8 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ suc 𝑚 ∧ 𝐶 ∈ 𝐴) → (𝐴 ∖ {𝐶}) ≈ 𝑚)
23	15, 20, 21, 22	syl3anc 1238	. . . . . . 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 6782	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐵 ≈ 𝐴)
27		entr 6783	. . . . . . . . . 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 6878	. . . . . . . . 9 ⊢ ((𝑚 ∈ ω ∧ 𝐵 ≈ suc 𝑚 ∧ 𝐷 ∈ 𝐵) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
32	15, 28, 30, 31	syl3anc 1238	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
33	32	ensymd 6782	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ≈ (𝐵 ∖ {𝐷}))
34		entr 6783	. . . . . . 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 2594	. . . 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 4603	. . . 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 798	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
42	3, 41	rexlimddv 2599	1 ⊢ (𝜑 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   = wceq 1353   ∈ wcel 2148  ∃wrex 2456   ∖ cdif 3126  ∅c0 3422  {csn 3592   class class class wbr 4003  suc csuc 4365  ωcom 4589   ≈ cen 6737  Fincfn 6739

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 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587

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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-er 6534  df-en 6740  df-fin 6742

This theorem is referenced by:  fisseneq  6930