Theorem dif1enen 7042

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 6912	. . 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 4087	. . . . . . 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 6947	. . . . 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 3497	. . . . . 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 623	. . 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 4087	. . . . . . . . . 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 7041	. . . . . . . 8 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ suc 𝑚 ∧ 𝐶 ∈ 𝐴) → (𝐴 ∖ {𝐶}) ≈ 𝑚)
23	15, 20, 21, 22	syl3anc 1271	. . . . . . 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 6935	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐵 ≈ 𝐴)
27		entr 6936	. . . . . . . . . 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 7041	. . . . . . . . 9 ⊢ ((𝑚 ∈ ω ∧ 𝐵 ≈ suc 𝑚 ∧ 𝐷 ∈ 𝐵) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
32	15, 28, 30, 31	syl3anc 1271	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
33	32	ensymd 6935	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ≈ (𝐵 ∖ {𝐷}))
34		entr 6936	. . . . . . 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 2648	. . . 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 4696	. . . 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 803	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
42	3, 41	rexlimddv 2653	1 ⊢ (𝜑 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   = wceq 1395   ∈ wcel 2200  ∃wrex 2509   ∖ cdif 3194  ∅c0 3491  {csn 3666   class class class wbr 4083  suc csuc 4456  ωcom 4682   ≈ cen 6885  Fincfn 6887

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-er 6680  df-en 6888  df-fin 6890

This theorem is referenced by:  fisseneq  7096