Theorem fiuneneq 43644

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 1143	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ∈ Fin)
2		enfi 9118	. . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
3	2	3ad2ant1 1139	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
4	1, 3	mpbid 233	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 ∈ Fin)
5		unfi 9102	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
6	1, 4, 5	syl2anc 590	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → (𝐴 ∪ 𝐵) ∈ Fin)
7		ssun1 4114	. . . . . 6 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
8	7	a1i 11	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ⊆ (𝐴 ∪ 𝐵))
9		simp3 1144	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → (𝐴 ∪ 𝐵) ≈ 𝐴)
10	9	ensymd 8949	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ≈ (𝐴 ∪ 𝐵))
11		fisseneq 9170	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ∈ Fin ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ 𝐴 ≈ (𝐴 ∪ 𝐵)) → 𝐴 = (𝐴 ∪ 𝐵))
12	6, 8, 10, 11	syl3anc 1379	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 = (𝐴 ∪ 𝐵))
13		ssun2 4115	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
14	13	a1i 11	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
15		simp1 1142	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 ≈ 𝐵)
16		entr 8950	. . . . . . 7 ⊢ (((𝐴 ∪ 𝐵) ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → (𝐴 ∪ 𝐵) ≈ 𝐵)
17	9, 15, 16	syl2anc 590	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → (𝐴 ∪ 𝐵) ≈ 𝐵)
18	17	ensymd 8949	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 ≈ (𝐴 ∪ 𝐵))
19		fisseneq 9170	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ∈ Fin ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ 𝐵 ≈ (𝐴 ∪ 𝐵)) → 𝐵 = (𝐴 ∪ 𝐵))
20	6, 14, 18, 19	syl3anc 1379	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐵 = (𝐴 ∪ 𝐵))
21	12, 20	eqtr4d 2778	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin ∧ (𝐴 ∪ 𝐵) ≈ 𝐴) → 𝐴 = 𝐵)
22	21	3expia 1127	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → ((𝐴 ∪ 𝐵) ≈ 𝐴 → 𝐴 = 𝐵))
23		enrefg 8928	. . . 4 ⊢ (𝐴 ∈ Fin → 𝐴 ≈ 𝐴)
24	23	adantl 482	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → 𝐴 ≈ 𝐴)
25		unidm 4094	. . . . 5 ⊢ (𝐴 ∪ 𝐴) = 𝐴
26		uneq2 4099	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 ∪ 𝐴) = (𝐴 ∪ 𝐵))
27	25, 26	eqtr3id 2789	. . . 4 ⊢ (𝐴 = 𝐵 → 𝐴 = (𝐴 ∪ 𝐵))
28	27	breq1d 5089	. . 3 ⊢ (𝐴 = 𝐵 → (𝐴 ≈ 𝐴 ↔ (𝐴 ∪ 𝐵) ≈ 𝐴))
29	24, 28	syl5ibcom 246	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → (𝐴 = 𝐵 → (𝐴 ∪ 𝐵) ≈ 𝐴))
30	22, 29	impbid 213	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐴 ∈ Fin) → ((𝐴 ∪ 𝐵) ≈ 𝐴 ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ∪ cun 3888   ⊆ wss 3890   class class class wbr 5079   ≈ cen 8887  Fincfn 8890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7814  df-1o 8402  df-er 8640  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894

This theorem is referenced by:  idomsubgmo  43645