Theorem dif1enen 7177

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 7040	. . 3 ⊢ (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
3	1, 2	sylib 122	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ω 𝐴 ≈ 𝑛)
4		simplrr 542	. . . . . 6 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ 𝑛)
5		breq2 4132	. . . . . . 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 7075	. . . . 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 3527	. . . . . 6 ⊢ (𝐶 ∈ 𝐴 → ¬ 𝐴 = ∅)
12	10, 11	syl 14	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 = ∅)
13	12	ad2antrr 492	. . . 4 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → ¬ 𝐴 = ∅)
14	9, 13	pm2.21dd 629	. . 3 ⊢ (((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑛 = ∅) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
15		simplr 533	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ∈ ω)
16		simprr 537	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → 𝐴 ≈ 𝑛)
17	16	ad2antrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴 ≈ 𝑛)
18		breq2 4132	. . . . . . . . . 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 496	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐶 ∈ 𝐴)
22		dif1en 7176	. . . . . . . 8 ⊢ ((𝑚 ∈ ω ∧ 𝐴 ≈ suc 𝑚 ∧ 𝐶 ∈ 𝐴) → (𝐴 ∖ {𝐶}) ≈ 𝑚)
23	15, 20, 21, 22	syl3anc 1278	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐴 ∖ {𝐶}) ≈ 𝑚)
24		dif1enen.ab	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ≈ 𝐵)
25	24	ad3antrrr 496	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴 ≈ 𝐵)
26	25	ensymd 7063	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐵 ≈ 𝐴)
27		entr 7064	. . . . . . . . . 10 ⊢ ((𝐵 ≈ 𝐴 ∧ 𝐴 ≈ suc 𝑚) → 𝐵 ≈ suc 𝑚)
28	26, 20, 27	syl2anc 415	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐵 ≈ suc 𝑚)
29		dif1enen.d	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ 𝐵)
30	29	ad3antrrr 496	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐷 ∈ 𝐵)
31		dif1en 7176	. . . . . . . . 9 ⊢ ((𝑚 ∈ ω ∧ 𝐵 ≈ suc 𝑚 ∧ 𝐷 ∈ 𝐵) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
32	15, 28, 30, 31	syl3anc 1278	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → (𝐵 ∖ {𝐷}) ≈ 𝑚)
33	32	ensymd 7063	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑚 ≈ (𝐵 ∖ {𝐷}))
34		entr 7064	. . . . . . 7 ⊢ (((𝐴 ∖ {𝐶}) ≈ 𝑚 ∧ 𝑚 ≈ (𝐵 ∖ {𝐷})) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
35	23, 33, 34	syl2anc 415	. . . . . 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 2668	. . . 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 4749	. . . 4 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
40	39	ad2antrl 494	. . 3 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
41	14, 38, 40	mpjaodan 810	. 2 ⊢ ((𝜑 ∧ (𝑛 ∈ ω ∧ 𝐴 ≈ 𝑛)) → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))
42	3, 41	rexlimddv 2673	1 ⊢ (𝜑 → (𝐴 ∖ {𝐶}) ≈ (𝐵 ∖ {𝐷}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402   ∈ wcel 2209  ∃wrex 2529   ∖ cdif 3217  ∅c0 3520  {csn 3708   class class class wbr 4128  suc csuc 4508  ωcom 4735   ≈ cen 7013  Fincfn 7015

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-er 6800  df-en 7016  df-fin 7018

This theorem is referenced by:  fisseneq  7235