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Theorem fiuneneq 43204
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 1138 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ∈ Fin)
2 enfi 9227 . . . . . . . 8 (𝐴𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
323ad2ant1 1134 . . . . . . 7 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
41, 3mpbid 232 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ∈ Fin)
5 unfi 9211 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
61, 4, 5syl2anc 584 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ∈ Fin)
7 ssun1 4178 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
87a1i 11 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ⊆ (𝐴𝐵))
9 simp3 1139 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ≈ 𝐴)
109ensymd 9045 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ≈ (𝐴𝐵))
11 fisseneq 9293 . . . . 5 (((𝐴𝐵) ∈ Fin ∧ 𝐴 ⊆ (𝐴𝐵) ∧ 𝐴 ≈ (𝐴𝐵)) → 𝐴 = (𝐴𝐵))
126, 8, 10, 11syl3anc 1373 . . . 4 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 = (𝐴𝐵))
13 ssun2 4179 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
1413a1i 11 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ⊆ (𝐴𝐵))
15 simp1 1137 . . . . . . 7 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴𝐵)
16 entr 9046 . . . . . . 7 (((𝐴𝐵) ≈ 𝐴𝐴𝐵) → (𝐴𝐵) ≈ 𝐵)
179, 15, 16syl2anc 584 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ≈ 𝐵)
1817ensymd 9045 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ≈ (𝐴𝐵))
19 fisseneq 9293 . . . . 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 1122 . 2 ((𝐴𝐵𝐴 ∈ Fin) → ((𝐴𝐵) ≈ 𝐴𝐴 = 𝐵))
23 enrefg 9024 . . . 4 (𝐴 ∈ Fin → 𝐴𝐴)
2423adantl 481 . . 3 ((𝐴𝐵𝐴 ∈ Fin) → 𝐴𝐴)
25 unidm 4157 . . . . 5 (𝐴𝐴) = 𝐴
26 uneq2 4162 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
2725, 26eqtr3id 2791 . . . 4 (𝐴 = 𝐵𝐴 = (𝐴𝐵))
2827breq1d 5153 . . 3 (𝐴 = 𝐵 → (𝐴𝐴 ↔ (𝐴𝐵) ≈ 𝐴))
2924, 28syl5ibcom 245 . 2 ((𝐴𝐵𝐴 ∈ Fin) → (𝐴 = 𝐵 → (𝐴𝐵) ≈ 𝐴))
3022, 29impbid 212 1 ((𝐴𝐵𝐴 ∈ Fin) → ((𝐴𝐵) ≈ 𝐴𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  cun 3949  wss 3951   class class class wbr 5143  cen 8982  Fincfn 8985
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989
This theorem is referenced by:  idomsubgmo  43205
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