Theorem fiuneneq 43181

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 1137	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ∈ Fin)
2		enfi 9151	. . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
3	2	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
4	1, 3	mpbid 232	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 ∈ Fin)
5		unfi 9135	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
6	1, 4, 5	syl2anc 584	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → (𝐴 ∪ 𝐵) ∈ Fin)
7		ssun1 4141	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
8	7	a1i 11	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ⊆ (𝐴 ∪ 𝐵))
9		simp3 1138	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → (𝐴 ∪ 𝐵) ≈ 𝐴)
10	9	ensymd 8976	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ≈ (𝐴 ∪ 𝐵))
11		fisseneq 9204	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ∈ Fin ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ 𝐴 ≈ (𝐴 ∪ 𝐵)) → 𝐴 = (𝐴 ∪ 𝐵))
12	6, 8, 10, 11	syl3anc 1373	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 = (𝐴 ∪ 𝐵))
13		ssun2 4142	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
14	13	a1i 11	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
15		simp1 1136	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ≈ 𝐵)
16		entr 8977	. . . . . . 7 ⊢ (((𝐴 ∪ 𝐵) ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → (𝐴 ∪ 𝐵) ≈ 𝐵)
17	9, 15, 16	syl2anc 584	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → (𝐴 ∪ 𝐵) ≈ 𝐵)
18	17	ensymd 8976	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 ≈ (𝐴 ∪ 𝐵))
19		fisseneq 9204	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ∈ Fin ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ 𝐵 ≈ (𝐴 ∪ 𝐵)) → 𝐵 = (𝐴 ∪ 𝐵))
20	6, 14, 18, 19	syl3anc 1373	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 = (𝐴 ∪ 𝐵))
21	12, 20	eqtr4d 2767	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 = 𝐵)
22	21	3expia 1121	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → ((𝐴 ∪ 𝐵) ≈ 𝐴 → 𝐴 = 𝐵))
23		enrefg 8955	. . . 4 ⊢ (𝐴 ∈ Fin → 𝐴 ≈ 𝐴)
24	23	adantl 481	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐴 ≈ 𝐴)
25		unidm 4120	. . . . 5 ⊢ (𝐴 ∪ 𝐴) = 𝐴
26		uneq2 4125	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐴) = (𝐴 ∪ 𝐵))
27	25, 26	eqtr3id 2778	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = (𝐴 ∪ 𝐵))
28	27	breq1d 5117	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≈ 𝐴 ↔ (𝐴 ∪ 𝐵) ≈ 𝐴))
29	24, 28	syl5ibcom 245	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → (𝐴 = 𝐵 → (𝐴 ∪ 𝐵) ≈ 𝐴))
30	22, 29	impbid 212	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → ((𝐴 ∪ 𝐵) ≈ 𝐴 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∪ cun 3912   ⊆ wss 3914   class class class wbr 5107   ≈ cen 8915  Fincfn 8918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-om 7843  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922

This theorem is referenced by:  idomsubgmo  43182