Theorem unfiOLD 9338

Description: Obsolete version of unfi 9197 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 9204	. 2 ⊢ (𝐵 ∈ Fin → (𝐵 ∖ 𝐴) ∈ Fin)
2		reeanv 3223	. . . 4 ⊢ (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) ↔ (∃𝑥 ∈ ω 𝐴 ≈ 𝑥 ∧ ∃𝑦 ∈ ω (𝐵 ∖ 𝐴) ≈ 𝑦))
3		isfi 8997	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴 ≈ 𝑥)
4		isfi 8997	. . . . 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 278	. . 3 ⊢ (∃𝑥 ∈ ω ∃𝑦 ∈ ω (𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) ↔ (𝐴 ∈ Fin ∧ (𝐵 ∖ 𝐴) ∈ Fin))
7		nnacl 8632	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝑥 +o 𝑦) ∈ ω)
8		unfilem3 9337	. . . . . . 7 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → 𝑦 ≈ ((𝑥 +o 𝑦) ∖ 𝑥))
9		entr 9027	. . . . . . . 8 ⊢ (((𝐵 ∖ 𝐴) ≈ 𝑦 ∧ 𝑦 ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) → (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥))
10	9	expcom 413	. . . . . . 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 4472	. . . . . . . 8 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐴)) = ∅
13		disjdif 4472	. . . . . . . 8 ⊢ (𝑥 ∩ ((𝑥 +o 𝑦) ∖ 𝑥)) = ∅
14		unen 9071	. . . . . . . 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 704	. . . . . . 7 ⊢ ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ ((𝑥 +o 𝑦) ∖ 𝑥)) → (𝐴 ∪ (𝐵 ∖ 𝐴)) ≈ (𝑥 ∪ ((𝑥 +o 𝑦) ∖ 𝑥)))
16		undif2 4477	. . . . . . . . 9 ⊢ (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵)
17	16	a1i 11	. . . . . . . 8 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → (𝐴 ∪ (𝐵 ∖ 𝐴)) = (𝐴 ∪ 𝐵))
18		nnaword1 8650	. . . . . . . . 9 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → 𝑥 ⊆ (𝑥 +o 𝑦))
19		undif 4482	. . . . . . . . 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 5161	. . . . . . 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 604	. . . . 5 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) → (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)))
24		breq2 5152	. . . . . . 7 ⊢ (𝑧 = (𝑥 +o 𝑦) → ((𝐴 ∪ 𝐵) ≈ 𝑧 ↔ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)))
25	24	rspcev 3609	. . . . . 6 ⊢ (((𝑥 +o 𝑦) ∈ ω ∧ (𝐴 ∪ 𝐵) ≈ (𝑥 +o 𝑦)) → ∃𝑧 ∈ ω (𝐴 ∪ 𝐵) ≈ 𝑧)
26		isfi 8997	. . . . . 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 683	. . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → ((𝐴 ≈ 𝑥 ∧ (𝐵 ∖ 𝐴) ≈ 𝑦) → (𝐴 ∪ 𝐵) ∈ Fin))
29	28	rexlimivv 3196	. . 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 592	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∃wrex 3067   ∖ cdif 3944   ∪ cun 3945   ∩ cin 3946   ⊆ wss 3947  ∅c0 4323   class class class wbr 5148  (class class class)co 7420  ωcom 7870   +_o coa 8484   ≈ cen 8961  Fincfn 8964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-2nd 7994  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-oadd 8491  df-er 8725  df-en 8965  df-fin 8968

This theorem is referenced by: (None)