Theorem unfidisj 7018

Description: The union of two disjoint finite sets is finite. (Contributed by Jim Kingdon, 25-Feb-2022.)

Assertion

Ref	Expression
unfidisj	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ Fin)

Proof of Theorem unfidisj

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uneq2 3320	. . 3 ⊢ (𝑤 = ∅ → (𝐴 ∪ 𝑤) = (𝐴 ∪ ∅))
2	1	eleq1d 2273	. 2 ⊢ (𝑤 = ∅ → ((𝐴 ∪ 𝑤) ∈ Fin ↔ (𝐴 ∪ ∅) ∈ Fin))
3		uneq2 3320	. . 3 ⊢ (𝑤 = 𝑦 → (𝐴 ∪ 𝑤) = (𝐴 ∪ 𝑦))
4	3	eleq1d 2273	. 2 ⊢ (𝑤 = 𝑦 → ((𝐴 ∪ 𝑤) ∈ Fin ↔ (𝐴 ∪ 𝑦) ∈ Fin))
5		uneq2 3320	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐴 ∪ 𝑤) = (𝐴 ∪ (𝑦 ∪ {𝑧})))
6	5	eleq1d 2273	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐴 ∪ 𝑤) ∈ Fin ↔ (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
7		uneq2 3320	. . 3 ⊢ (𝑤 = 𝐵 → (𝐴 ∪ 𝑤) = (𝐴 ∪ 𝐵))
8	7	eleq1d 2273	. 2 ⊢ (𝑤 = 𝐵 → ((𝐴 ∪ 𝑤) ∈ Fin ↔ (𝐴 ∪ 𝐵) ∈ Fin))
9		un0 3493	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
10		simp1 999	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ Fin)
11	9, 10	eqeltrid 2291	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ ∅) ∈ Fin)
12		unass 3329	. . . 4 ⊢ ((𝐴 ∪ 𝑦) ∪ {𝑧}) = (𝐴 ∪ (𝑦 ∪ {𝑧}))
13		simpr 110	. . . . 5 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∪ 𝑦) ∈ Fin)
14		vex 2774	. . . . . 6 ⊢ 𝑧 ∈ V
15	14	a1i 9	. . . . 5 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → 𝑧 ∈ V)
16		simplrr 536	. . . . . . . . 9 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → 𝑧 ∈ (𝐵 ∖ 𝑦))
17	16	eldifad 3176	. . . . . . . 8 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → 𝑧 ∈ 𝐵)
18		simp3 1001	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
19	18	ad3antrrr 492	. . . . . . . 8 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∩ 𝐵) = ∅)
20		minel 3521	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ¬ 𝑧 ∈ 𝐴)
21	17, 19, 20	syl2anc 411	. . . . . . 7 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → ¬ 𝑧 ∈ 𝐴)
22	16	eldifbd 3177	. . . . . . 7 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → ¬ 𝑧 ∈ 𝑦)
23		ioran 753	. . . . . . 7 ⊢ (¬ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦) ↔ (¬ 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ 𝑦))
24	21, 22, 23	sylanbrc 417	. . . . . 6 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → ¬ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦))
25		elun 3313	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∪ 𝑦) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦))
26	24, 25	sylnibr 678	. . . . 5 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → ¬ 𝑧 ∈ (𝐴 ∪ 𝑦))
27		unsnfi 7015	. . . . 5 ⊢ (((𝐴 ∪ 𝑦) ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ (𝐴 ∪ 𝑦)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ∈ Fin)
28	13, 15, 26, 27	syl3anc 1249	. . . 4 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ∈ Fin)
29	12, 28	eqeltrrid 2292	. . 3 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin)
30	29	ex 115	. 2 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) → ((𝐴 ∪ 𝑦) ∈ Fin → (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
31		simp2 1000	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ Fin)
32	2, 4, 6, 8, 11, 30, 31	findcard2sd 6988	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709   ∧ w3a 980   = wceq 1372   ∈ wcel 2175  Vcvv 2771   ∖ cdif 3162   ∪ cun 3163   ∩ cin 3164   ⊆ wss 3165  ∅c0 3459  {csn 3632  Fincfn 6826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-1o 6501  df-er 6619  df-en 6827  df-fin 6829

This theorem is referenced by:  unfiin  7022  prfidisj  7023  tpfidisj  7025  xpfi  7028  iunfidisj  7047  hashunlem  10947  hashun  10948  fsumsplitsnun  11672  fsum2dlemstep  11687  fsumconst  11707  fprodsplitsn  11886