Theorem unfiOLD 9309

Description: Obsolete version of unfi 9168 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 9175	. 2 ⊢ (𝐵 ∈ Fin → (𝐵 ∖ 𝐴) ∈ Fin)
2		reeanv 3226	. . . 4 ⊢ (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) ↔ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ∧ ∃𝑦 ∈ ω (𝐵 ∖ 𝐴) ≈ 𝑦))
3		isfi 8968	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		isfi 8968	. . . . 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 8607	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 +o 𝑦) ∈ ω)
8		unfilem3 9308	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → 𝑦 ≈ ((𝑥 +o 𝑦) ∖ 𝑥))
9		entr 8998	. . . . . . . 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 4470	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
13		disjdif 4470	. . . . . . . 8 ⊢ (𝑥 ∩ ((𝑥 +o 𝑦) ∖ 𝑥)) = ∅
14		unen 9042	. . . . . . . 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 703	. . . . . . 7 ⊢ ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) → (𝐴 ∪ (𝐵 ∖ 𝐴)) ≈ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)))
16		undif2 4475	. . . . . . . . 9 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
17	16	a1i 11	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵))
18		nnaword1 8625	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → 𝑥 ⊆ (𝑥 +o 𝑦))
19		undif 4480	. . . . . . . . 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 5160	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ∪ (𝐵 ∖ 𝐴)) ≈ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)) ↔ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)))
22	15, 21	imbitrid 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 5151	. . . . . . 7 ⊢ (𝑧 = (𝑥 +o 𝑦) → ((𝐴 ∪ 𝐵) ≈ 𝑧 ↔ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)))
25	24	rspcev 3612	. . . . . 6 ⊢ (((𝑥 +o 𝑦) ∈ ω ∧ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)) → ∃𝑧 ∈ ω (𝐴 ∪ 𝐵) ≈ 𝑧)
26		isfi 8968	. . . . . 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 682	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) → (𝐴 ∪ 𝐵) ∈ Fin))
29	28	rexlimivv 3199	. . 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 1541   ∈ wcel 2106  ∃wrex 3070   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321   class class class wbr 5147  (class class class)co 7405  ωcom 7851   +_o coa 8459   ≈ cen 8932  Fincfn 8935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-oadd 8466  df-er 8699  df-en 8936  df-fin 8939

This theorem is referenced by: (None)