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Theorem fiuneneq 37295
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 1060 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ∈ Fin)
2 enfi 8136 . . . . . . . 8 (𝐴𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
323ad2ant1 1080 . . . . . . 7 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
41, 3mpbid 222 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ∈ Fin)
5 unfi 8187 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
61, 4, 5syl2anc 692 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ∈ Fin)
7 ssun1 3760 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
87a1i 11 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ⊆ (𝐴𝐵))
9 simp3 1061 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ≈ 𝐴)
109ensymd 7967 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ≈ (𝐴𝐵))
11 fisseneq 8131 . . . . 5 (((𝐴𝐵) ∈ Fin ∧ 𝐴 ⊆ (𝐴𝐵) ∧ 𝐴 ≈ (𝐴𝐵)) → 𝐴 = (𝐴𝐵))
126, 8, 10, 11syl3anc 1323 . . . 4 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 = (𝐴𝐵))
13 ssun2 3761 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
1413a1i 11 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ⊆ (𝐴𝐵))
15 simp1 1059 . . . . . . 7 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴𝐵)
16 entr 7968 . . . . . . 7 (((𝐴𝐵) ≈ 𝐴𝐴𝐵) → (𝐴𝐵) ≈ 𝐵)
179, 15, 16syl2anc 692 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ≈ 𝐵)
1817ensymd 7967 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ≈ (𝐴𝐵))
19 fisseneq 8131 . . . . 5 (((𝐴𝐵) ∈ Fin ∧ 𝐵 ⊆ (𝐴𝐵) ∧ 𝐵 ≈ (𝐴𝐵)) → 𝐵 = (𝐴𝐵))
206, 14, 18, 19syl3anc 1323 . . . 4 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 = (𝐴𝐵))
2112, 20eqtr4d 2658 . . 3 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 = 𝐵)
22213expia 1264 . 2 ((𝐴𝐵𝐴 ∈ Fin) → ((𝐴𝐵) ≈ 𝐴𝐴 = 𝐵))
23 enrefg 7947 . . . 4 (𝐴 ∈ Fin → 𝐴𝐴)
2423adantl 482 . . 3 ((𝐴𝐵𝐴 ∈ Fin) → 𝐴𝐴)
25 unidm 3740 . . . . 5 (𝐴𝐴) = 𝐴
26 uneq2 3745 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
2725, 26syl5eqr 2669 . . . 4 (𝐴 = 𝐵𝐴 = (𝐴𝐵))
2827breq1d 4633 . . 3 (𝐴 = 𝐵 → (𝐴𝐴 ↔ (𝐴𝐵) ≈ 𝐴))
2924, 28syl5ibcom 235 . 2 ((𝐴𝐵𝐴 ∈ Fin) → (𝐴 = 𝐵 → (𝐴𝐵) ≈ 𝐴))
3022, 29impbid 202 1 ((𝐴𝐵𝐴 ∈ Fin) → ((𝐴𝐵) ≈ 𝐴𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  cun 3558  wss 3560   class class class wbr 4623  cen 7912  Fincfn 7915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-oadd 7524  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919
This theorem is referenced by:  idomsubgmo  37296
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