Theorem fiuneneq 43960

Description: Two finite sets of equal size have a union of the same size iff they were equal. (Contributed by Stefan O'Rear, 12-Sep-2015.)

Assertion

Ref	Expression
fiuneneq	⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → ((𝐴 ∪ 𝐵) ≈ 𝐴 ↔ 𝐴 = 𝐵))

Proof of Theorem fiuneneq

Step	Hyp	Ref	Expression
1		simp2 1155	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ∈ Fin)
2		enfi 9181	. . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
3	2	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
4	1, 3	mpbid 235	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 ∈ Fin)
5		unfi 9165	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
6	1, 4, 5	syl2anc 596	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → (𝐴 ∪ 𝐵) ∈ Fin)
7		ssun1 4134	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
8	7	a1i 11	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ⊆ (𝐴 ∪ 𝐵))
9		simp3 1156	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → (𝐴 ∪ 𝐵) ≈ 𝐴)
10	9	ensymd 9011	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ≈ (𝐴 ∪ 𝐵))
11		fisseneq 9233	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ∈ Fin ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ 𝐴 ≈ (𝐴 ∪ 𝐵)) → 𝐴 = (𝐴 ∪ 𝐵))
12	6, 8, 10, 11	syl3anc 1398	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 = (𝐴 ∪ 𝐵))
13		ssun2 4135	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
14	13	a1i 11	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
15		simp1 1154	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ≈ 𝐵)
16		entr 9012	. . . . . . 7 ⊢ (((𝐴 ∪ 𝐵) ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → (𝐴 ∪ 𝐵) ≈ 𝐵)
17	9, 15, 16	syl2anc 596	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → (𝐴 ∪ 𝐵) ≈ 𝐵)
18	17	ensymd 9011	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 ≈ (𝐴 ∪ 𝐵))
19		fisseneq 9233	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ∈ Fin ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ 𝐵 ≈ (𝐴 ∪ 𝐵)) → 𝐵 = (𝐴 ∪ 𝐵))
20	6, 14, 18, 19	syl3anc 1398	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 = (𝐴 ∪ 𝐵))
21	12, 20	eqtr4d 2804	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 = 𝐵)
22	21	3expia 1139	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → ((𝐴 ∪ 𝐵) ≈ 𝐴 → 𝐴 = 𝐵))
23		enrefg 8990	. . . 4 ⊢ (𝐴 ∈ Fin → 𝐴 ≈ 𝐴)
24	23	adantl 487	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐴 ≈ 𝐴)
25		unidm 4114	. . . . 5 ⊢ (𝐴 ∪ 𝐴) = 𝐴
26		uneq2 4119	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐴) = (𝐴 ∪ 𝐵))
27	25, 26	eqtr3id 2815	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = (𝐴 ∪ 𝐵))
28	27	breq1d 5124	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≈ 𝐴 ↔ (𝐴 ∪ 𝐵) ≈ 𝐴))
29	24, 28	syl5ibcom 248	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → (𝐴 = 𝐵 → (𝐴 ∪ 𝐵) ≈ 𝐴))
30	22, 29	impbid 215	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → ((𝐴 ∪ 𝐵) ≈ 𝐴 ↔ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ∪ cun 3906   ⊆ wss 3908   class class class wbr 5114   ≈ cen 8949  Fincfn 8952

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ord 6370  df-on 6371  df-lim 6372  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7872  df-1o 8462  df-er 8703  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956

This theorem is used by:  idomsubgmo  43961