Theorem unfidisj 7113

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 3355	. . 3 ⊢ (𝑤 = ∅ → (𝐴 ∪ 𝑤) = (𝐴 ∪ ∅))
2	1	eleq1d 2300	. 2 ⊢ (𝑤 = ∅ → ((𝐴 ∪ 𝑤) ∈ Fin ↔ (𝐴 ∪ ∅) ∈ Fin))
3		uneq2 3355	. . 3 ⊢ (𝑤 = 𝑦 → (𝐴 ∪ 𝑤) = (𝐴 ∪ 𝑦))
4	3	eleq1d 2300	. 2 ⊢ (𝑤 = 𝑦 → ((𝐴 ∪ 𝑤) ∈ Fin ↔ (𝐴 ∪ 𝑦) ∈ Fin))
5		uneq2 3355	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐴 ∪ 𝑤) = (𝐴 ∪ (𝑦 ∪ {𝑧})))
6	5	eleq1d 2300	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐴 ∪ 𝑤) ∈ Fin ↔ (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
7		uneq2 3355	. . 3 ⊢ (𝑤 = 𝐵 → (𝐴 ∪ 𝑤) = (𝐴 ∪ 𝐵))
8	7	eleq1d 2300	. 2 ⊢ (𝑤 = 𝐵 → ((𝐴 ∪ 𝑤) ∈ Fin ↔ (𝐴 ∪ 𝐵) ∈ Fin))
9		un0 3528	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
10		simp1 1023	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ Fin)
11	9, 10	eqeltrid 2318	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ ∅) ∈ Fin)
12		unass 3364	. . . 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 2805	. . . . . 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 538	. . . . . . . . 9 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → 𝑧 ∈ (𝐵 ∖ 𝑦))
17	16	eldifad 3211	. . . . . . . 8 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → 𝑧 ∈ 𝐵)
18		simp3 1025	. . . . . . . . 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 3556	. . . . . . . 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 3212	. . . . . . 7 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → ¬ 𝑧 ∈ 𝑦)
23		ioran 759	. . . . . . 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 3348	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∪ 𝑦) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦))
26	24, 25	sylnibr 683	. . . . 5 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → ¬ 𝑧 ∈ (𝐴 ∪ 𝑦))
27		unsnfi 7110	. . . . 5 ⊢ (((𝐴 ∪ 𝑦) ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ (𝐴 ∪ 𝑦)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ∈ Fin)
28	13, 15, 26, 27	syl3anc 1273	. . . 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 2319	. . 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 1024	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ Fin)
32	2, 4, 6, 8, 11, 30, 31	findcard2sd 7080	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 715   ∧ w3a 1004   = wceq 1397   ∈ wcel 2202  Vcvv 2802   ∖ cdif 3197   ∪ cun 3198   ∩ cin 3199   ⊆ wss 3200  ∅c0 3494  {csn 3669  Fincfn 6908

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6581  df-er 6701  df-en 6909  df-fin 6911

This theorem is referenced by:  unfiin  7117  prfidisj  7118  tpfidisj  7120  xpfi  7123  iunfidisj  7144  hashunlem  11066  hashun  11067  fsumsplitsnun  11979  fsum2dlemstep  11994  fsumconst  12014  fprodsplitsn  12193  vtxdfifiun  16147