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Theorem fiuneneq 40725
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 ((𝐴𝐵𝐴 ∈ Fin) → ((𝐴𝐵) ≈ 𝐴𝐴 = 𝐵))

Proof of Theorem fiuneneq
StepHypRef Expression
1 simp2 1139 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ∈ Fin)
2 enfi 8865 . . . . . . . 8 (𝐴𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
323ad2ant1 1135 . . . . . . 7 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
41, 3mpbid 235 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ∈ Fin)
5 unfi 8850 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
61, 4, 5syl2anc 587 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ∈ Fin)
7 ssun1 4086 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
87a1i 11 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ⊆ (𝐴𝐵))
9 simp3 1140 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ≈ 𝐴)
109ensymd 8679 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ≈ (𝐴𝐵))
11 fisseneq 8889 . . . . 5 (((𝐴𝐵) ∈ Fin ∧ 𝐴 ⊆ (𝐴𝐵) ∧ 𝐴 ≈ (𝐴𝐵)) → 𝐴 = (𝐴𝐵))
126, 8, 10, 11syl3anc 1373 . . . 4 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 = (𝐴𝐵))
13 ssun2 4087 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
1413a1i 11 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ⊆ (𝐴𝐵))
15 simp1 1138 . . . . . . 7 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴𝐵)
16 entr 8680 . . . . . . 7 (((𝐴𝐵) ≈ 𝐴𝐴𝐵) → (𝐴𝐵) ≈ 𝐵)
179, 15, 16syl2anc 587 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ≈ 𝐵)
1817ensymd 8679 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ≈ (𝐴𝐵))
19 fisseneq 8889 . . . . 5 (((𝐴𝐵) ∈ Fin ∧ 𝐵 ⊆ (𝐴𝐵) ∧ 𝐵 ≈ (𝐴𝐵)) → 𝐵 = (𝐴𝐵))
206, 14, 18, 19syl3anc 1373 . . . 4 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 = (𝐴𝐵))
2112, 20eqtr4d 2780 . . 3 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 = 𝐵)
22213expia 1123 . 2 ((𝐴𝐵𝐴 ∈ Fin) → ((𝐴𝐵) ≈ 𝐴𝐴 = 𝐵))
23 enrefg 8660 . . . 4 (𝐴 ∈ Fin → 𝐴𝐴)
2423adantl 485 . . 3 ((𝐴𝐵𝐴 ∈ Fin) → 𝐴𝐴)
25 unidm 4066 . . . . 5 (𝐴𝐴) = 𝐴
26 uneq2 4071 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
2725, 26eqtr3id 2792 . . . 4 (𝐴 = 𝐵𝐴 = (𝐴𝐵))
2827breq1d 5063 . . 3 (𝐴 = 𝐵 → (𝐴𝐴 ↔ (𝐴𝐵) ≈ 𝐴))
2924, 28syl5ibcom 248 . 2 ((𝐴𝐵𝐴 ∈ Fin) → (𝐴 = 𝐵 → (𝐴𝐵) ≈ 𝐴))
3022, 29impbid 215 1 ((𝐴𝐵𝐴 ∈ Fin) → ((𝐴𝐵) ≈ 𝐴𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  cun 3864  wss 3866   class class class wbr 5053  cen 8623  Fincfn 8626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-om 7645  df-1o 8202  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630
This theorem is referenced by:  idomsubgmo  40726
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