Theorem fiuneneq 42241

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 9192	. . . . . . . 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 231	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 ∈ Fin)
5		unfi 9174	. . . . . 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 4172	. . . . . 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 9003	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ≈ (𝐴 ∪ 𝐵))
11		fisseneq 9259	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ∈ Fin ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ 𝐴 ≈ (𝐴 ∪ 𝐵)) → 𝐴 = (𝐴 ∪ 𝐵))
12	6, 8, 10, 11	syl3anc 1371	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 = (𝐴 ∪ 𝐵))
13		ssun2 4173	. . . . . 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 9004	. . . . . . 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 9003	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 ≈ (𝐴 ∪ 𝐵))
19		fisseneq 9259	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ∈ Fin ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ 𝐵 ≈ (𝐴 ∪ 𝐵)) → 𝐵 = (𝐴 ∪ 𝐵))
20	6, 14, 18, 19	syl3anc 1371	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 = (𝐴 ∪ 𝐵))
21	12, 20	eqtr4d 2775	. . 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 8982	. . . 4 ⊢ (𝐴 ∈ Fin → 𝐴 ≈ 𝐴)
24	23	adantl 482	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐴 ≈ 𝐴)
25		unidm 4152	. . . . 5 ⊢ (𝐴 ∪ 𝐴) = 𝐴
26		uneq2 4157	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐴) = (𝐴 ∪ 𝐵))
27	25, 26	eqtr3id 2786	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = (𝐴 ∪ 𝐵))
28	27	breq1d 5158	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≈ 𝐴 ↔ (𝐴 ∪ 𝐵) ≈ 𝐴))
29	24, 28	syl5ibcom 244	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → (𝐴 = 𝐵 → (𝐴 ∪ 𝐵) ≈ 𝐴))
30	22, 29	impbid 211	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → ((𝐴 ∪ 𝐵) ≈ 𝐴 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∪ cun 3946   ⊆ wss 3948   class class class wbr 5148   ≈ cen 8938  Fincfn 8941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7858  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945

This theorem is referenced by:  idomsubgmo  42242