Theorem unfiOLD 9081

Description: Obsolete version of unfi 8955 as of 7-Aug-2024. (Contributed by NM, 16-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
unfiOLD	⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)

Proof of Theorem unfiOLD

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

Step	Hyp	Ref	Expression
1		diffi 8962	. 2 ⊢ (𝐵 ∈ Fin → (𝐵 ∖ 𝐴) ∈ Fin)
2		reeanv 3294	. . . 4 ⊢ (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) ↔ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ∧ ∃𝑦 ∈ ω (𝐵 ∖ 𝐴) ≈ 𝑦))
3		isfi 8764	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		isfi 8764	. . . . 5 ⊢ ((𝐵 ∖ 𝐴) ∈ Fin ↔ ∃𝑦 ∈ ω (𝐵 ∖ 𝐴) ≈ 𝑦)
5	3, 4	anbi12i 627	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∖ 𝐴) ∈ Fin) ↔ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ∧ ∃𝑦 ∈ ω (𝐵 ∖ 𝐴) ≈ 𝑦))
6	2, 5	bitr4i 277	. . 3 ⊢ (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) ↔ (𝐴 ∈ Fin ∧ (𝐵 ∖ 𝐴) ∈ Fin))
7		nnacl 8442	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 +o 𝑦) ∈ ω)
8		unfilem3 9080	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → 𝑦 ≈ ((𝑥 +o 𝑦) ∖ 𝑥))
9		entr 8792	. . . . . . . 8 ⊢ (((𝐵 ∖ 𝐴) ≈ 𝑦 ∧ 𝑦 ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) → (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥))
10	9	expcom 414	. . . . . . 7 ⊢ (𝑦 ≈ ((𝑥 +o 𝑦) ∖ 𝑥) → ((𝐵 ∖ 𝐴) ≈ 𝑦 → (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)))
11	8, 10	syl 17	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐵 ∖ 𝐴) ≈ 𝑦 → (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)))
12		disjdif 4405	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
13		disjdif 4405	. . . . . . . 8 ⊢ (𝑥 ∩ ((𝑥 +o 𝑦) ∖ 𝑥)) = ∅
14		unen 8836	. . . . . . . 8 ⊢ (((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) ∧ ((𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅ ∧ (𝑥 ∩ ((𝑥 +o 𝑦) ∖ 𝑥)) = ∅)) → (𝐴 ∪ (𝐵 ∖ 𝐴)) ≈ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)))
15	12, 13, 14	mpanr12 702	. . . . . . 7 ⊢ ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) → (𝐴 ∪ (𝐵 ∖ 𝐴)) ≈ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)))
16		undif2 4410	. . . . . . . . 9 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
17	16	a1i 11	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵))
18		nnaword1 8460	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → 𝑥 ⊆ (𝑥 +o 𝑦))
19		undif 4415	. . . . . . . . 9 ⊢ (𝑥 ⊆ (𝑥 +o 𝑦) ↔ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)) = (𝑥 +o 𝑦))
20	18, 19	sylib 217	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)) = (𝑥 +o 𝑦))
21	17, 20	breq12d 5087	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ∪ (𝐵 ∖ 𝐴)) ≈ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)) ↔ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)))
22	15, 21	syl5ib 243	. . . . . 6 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) → (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)))
23	11, 22	sylan2d 605	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) → (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)))
24		breq2 5078	. . . . . . 7 ⊢ (𝑧 = (𝑥 +o 𝑦) → ((𝐴 ∪ 𝐵) ≈ 𝑧 ↔ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)))
25	24	rspcev 3561	. . . . . 6 ⊢ (((𝑥 +o 𝑦) ∈ ω ∧ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)) → ∃𝑧 ∈ ω (𝐴 ∪ 𝐵) ≈ 𝑧)
26		isfi 8764	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin ↔ ∃𝑧 ∈ ω (𝐴 ∪ 𝐵) ≈ 𝑧)
27	25, 26	sylibr 233	. . . . 5 ⊢ (((𝑥 +o 𝑦) ∈ ω ∧ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)) → (𝐴 ∪ 𝐵) ∈ Fin)
28	7, 23, 27	syl6an 681	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) → (𝐴 ∪ 𝐵) ∈ Fin))
29	28	rexlimivv 3221	. . 3 ⊢ (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) → (𝐴 ∪ 𝐵) ∈ Fin)
30	6, 29	sylbir 234	. 2 ⊢ ((𝐴 ∈ Fin ∧ (𝐵 ∖ 𝐴) ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
31	1, 30	sylan2 593	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∃wrex 3065   ∖ cdif 3884   ∪ cun 3885   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256   class class class wbr 5074  (class class class)co 7275  ωcom 7712   +_o coa 8294   ≈ cen 8730  Fincfn 8733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-er 8498  df-en 8734  df-fin 8737

This theorem is referenced by: (None)