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Theorem fiuneneq 36794
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 1055 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ∈ Fin)
2 enfi 8061 . . . . . . . 8 (𝐴𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
323ad2ant1 1075 . . . . . . 7 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
41, 3mpbid 221 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ∈ Fin)
5 unfi 8112 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
61, 4, 5syl2anc 691 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ∈ Fin)
7 ssun1 3738 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
87a1i 11 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ⊆ (𝐴𝐵))
9 simp3 1056 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ≈ 𝐴)
109ensymd 7893 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ≈ (𝐴𝐵))
11 fisseneq 8056 . . . . 5 (((𝐴𝐵) ∈ Fin ∧ 𝐴 ⊆ (𝐴𝐵) ∧ 𝐴 ≈ (𝐴𝐵)) → 𝐴 = (𝐴𝐵))
126, 8, 10, 11syl3anc 1318 . . . 4 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 = (𝐴𝐵))
13 ssun2 3739 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
1413a1i 11 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ⊆ (𝐴𝐵))
15 simp1 1054 . . . . . . 7 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴𝐵)
16 entr 7894 . . . . . . 7 (((𝐴𝐵) ≈ 𝐴𝐴𝐵) → (𝐴𝐵) ≈ 𝐵)
179, 15, 16syl2anc 691 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ≈ 𝐵)
1817ensymd 7893 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ≈ (𝐴𝐵))
19 fisseneq 8056 . . . . 5 (((𝐴𝐵) ∈ Fin ∧ 𝐵 ⊆ (𝐴𝐵) ∧ 𝐵 ≈ (𝐴𝐵)) → 𝐵 = (𝐴𝐵))
206, 14, 18, 19syl3anc 1318 . . . 4 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 = (𝐴𝐵))
2112, 20eqtr4d 2647 . . 3 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 = 𝐵)
22213expia 1259 . 2 ((𝐴𝐵𝐴 ∈ Fin) → ((𝐴𝐵) ≈ 𝐴𝐴 = 𝐵))
23 enrefg 7873 . . . 4 (𝐴 ∈ Fin → 𝐴𝐴)
2423adantl 481 . . 3 ((𝐴𝐵𝐴 ∈ Fin) → 𝐴𝐴)
25 unidm 3718 . . . . 5 (𝐴𝐴) = 𝐴
26 uneq2 3723 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
2725, 26syl5eqr 2658 . . . 4 (𝐴 = 𝐵𝐴 = (𝐴𝐵))
2827breq1d 4593 . . 3 (𝐴 = 𝐵 → (𝐴𝐴 ↔ (𝐴𝐵) ≈ 𝐴))
2924, 28syl5ibcom 234 . 2 ((𝐴𝐵𝐴 ∈ Fin) → (𝐴 = 𝐵 → (𝐴𝐵) ≈ 𝐴))
3022, 29impbid 201 1 ((𝐴𝐵𝐴 ∈ Fin) → ((𝐴𝐵) ≈ 𝐴𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  cun 3538  wss 3540   class class class wbr 4583  cen 7838  Fincfn 7841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845
This theorem is referenced by:  idomsubgmo  36795
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