Theorem unfidisj 7045

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 3329	. . 3 ⊢ (𝑤 = ∅ → (𝐴 ∪ 𝑤) = (𝐴 ∪ ∅))
2	1	eleq1d 2276	. 2 ⊢ (𝑤 = ∅ → ((𝐴 ∪ 𝑤) ∈ Fin ↔ (𝐴 ∪ ∅) ∈ Fin))
3		uneq2 3329	. . 3 ⊢ (𝑤 = 𝑦 → (𝐴 ∪ 𝑤) = (𝐴 ∪ 𝑦))
4	3	eleq1d 2276	. 2 ⊢ (𝑤 = 𝑦 → ((𝐴 ∪ 𝑤) ∈ Fin ↔ (𝐴 ∪ 𝑦) ∈ Fin))
5		uneq2 3329	. . 3 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → (𝐴 ∪ 𝑤) = (𝐴 ∪ (𝑦 ∪ {𝑧})))
6	5	eleq1d 2276	. 2 ⊢ (𝑤 = (𝑦 ∪ {𝑧}) → ((𝐴 ∪ 𝑤) ∈ Fin ↔ (𝐴 ∪ (𝑦 ∪ {𝑧})) ∈ Fin))
7		uneq2 3329	. . 3 ⊢ (𝑤 = 𝐵 → (𝐴 ∪ 𝑤) = (𝐴 ∪ 𝐵))
8	7	eleq1d 2276	. 2 ⊢ (𝑤 = 𝐵 → ((𝐴 ∪ 𝑤) ∈ Fin ↔ (𝐴 ∪ 𝐵) ∈ Fin))
9		un0 3502	. . 3 ⊢ (𝐴 ∪ ∅) = 𝐴
10		simp1 1000	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ Fin)
11	9, 10	eqeltrid 2294	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ ∅) ∈ Fin)
12		unass 3338	. . . 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 2779	. . . . . 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 3185	. . . . . . . 8 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → 𝑧 ∈ 𝐵)
18		simp3 1002	. . . . . . . . 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 3530	. . . . . . . 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 3186	. . . . . . 7 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → ¬ 𝑧 ∈ 𝑦)
23		ioran 754	. . . . . . 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 3322	. . . . . 6 ⊢ (𝑧 ∈ (𝐴 ∪ 𝑦) ↔ (𝑧 ∈ 𝐴 ∨ 𝑧 ∈ 𝑦))
26	24, 25	sylnibr 679	. . . . 5 ⊢ (((((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑦 ∈ Fin) ∧ (𝑦 ⊆ 𝐵 ∧ 𝑧 ∈ (𝐵 ∖ 𝑦))) ∧ (𝐴 ∪ 𝑦) ∈ Fin) → ¬ 𝑧 ∈ (𝐴 ∪ 𝑦))
27		unsnfi 7042	. . . . 5 ⊢ (((𝐴 ∪ 𝑦) ∈ Fin ∧ 𝑧 ∈ V ∧ ¬ 𝑧 ∈ (𝐴 ∪ 𝑦)) → ((𝐴 ∪ 𝑦) ∪ {𝑧}) ∈ Fin)
28	13, 15, 26, 27	syl3anc 1250	. . . 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 2295	. . 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 1001	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ Fin)
32	2, 4, 6, 8, 11, 30, 31	findcard2sd 7015	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ Fin)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 710   ∧ w3a 981   = wceq 1373   ∈ wcel 2178  Vcvv 2776   ∖ cdif 3171   ∪ cun 3172   ∩ cin 3173   ⊆ wss 3174  ∅c0 3468  {csn 3643  Fincfn 6850

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-1o 6525  df-er 6643  df-en 6851  df-fin 6853

This theorem is referenced by:  unfiin  7049  prfidisj  7050  tpfidisj  7052  xpfi  7055  iunfidisj  7074  hashunlem  10986  hashun  10987  fsumsplitsnun  11845  fsum2dlemstep  11860  fsumconst  11880  fprodsplitsn  12059