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Theorem fiuneneq 43180
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 1136 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ∈ Fin)
2 enfi 9224 . . . . . . . 8 (𝐴𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
323ad2ant1 1132 . . . . . . 7 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
41, 3mpbid 232 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ∈ Fin)
5 unfi 9209 . . . . . 6 ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴𝐵) ∈ Fin)
61, 4, 5syl2anc 584 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ∈ Fin)
7 ssun1 4187 . . . . . 6 𝐴 ⊆ (𝐴𝐵)
87a1i 11 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ⊆ (𝐴𝐵))
9 simp3 1137 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ≈ 𝐴)
109ensymd 9043 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 ≈ (𝐴𝐵))
11 fisseneq 9290 . . . . 5 (((𝐴𝐵) ∈ Fin ∧ 𝐴 ⊆ (𝐴𝐵) ∧ 𝐴 ≈ (𝐴𝐵)) → 𝐴 = (𝐴𝐵))
126, 8, 10, 11syl3anc 1370 . . . 4 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 = (𝐴𝐵))
13 ssun2 4188 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
1413a1i 11 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ⊆ (𝐴𝐵))
15 simp1 1135 . . . . . . 7 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴𝐵)
16 entr 9044 . . . . . . 7 (((𝐴𝐵) ≈ 𝐴𝐴𝐵) → (𝐴𝐵) ≈ 𝐵)
179, 15, 16syl2anc 584 . . . . . 6 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → (𝐴𝐵) ≈ 𝐵)
1817ensymd 9043 . . . . 5 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 ≈ (𝐴𝐵))
19 fisseneq 9290 . . . . 5 (((𝐴𝐵) ∈ Fin ∧ 𝐵 ⊆ (𝐴𝐵) ∧ 𝐵 ≈ (𝐴𝐵)) → 𝐵 = (𝐴𝐵))
206, 14, 18, 19syl3anc 1370 . . . 4 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐵 = (𝐴𝐵))
2112, 20eqtr4d 2777 . . 3 ((𝐴𝐵𝐴 ∈ Fin ∧ (𝐴𝐵) ≈ 𝐴) → 𝐴 = 𝐵)
22213expia 1120 . 2 ((𝐴𝐵𝐴 ∈ Fin) → ((𝐴𝐵) ≈ 𝐴𝐴 = 𝐵))
23 enrefg 9022 . . . 4 (𝐴 ∈ Fin → 𝐴𝐴)
2423adantl 481 . . 3 ((𝐴𝐵𝐴 ∈ Fin) → 𝐴𝐴)
25 unidm 4166 . . . . 5 (𝐴𝐴) = 𝐴
26 uneq2 4171 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐴) = (𝐴𝐵))
2725, 26eqtr3id 2788 . . . 4 (𝐴 = 𝐵𝐴 = (𝐴𝐵))
2827breq1d 5157 . . 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 1086   = wceq 1536  wcel 2105  cun 3960  wss 3962   class class class wbr 5147  cen 8980  Fincfn 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987
This theorem is referenced by:  idomsubgmo  43181
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