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Theorem difinf 9218

Description: An infinite set 𝐴 minus a finite set is infinite. (Contributed by FL, 3-Aug-2009.)

**Assertion**
Ref	Expression
difinf	⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ¬ (𝐴 ∖ 𝐵) ∈ Fin)

**Proof of Theorem difinf**
Step	Hyp	Ref	Expression
1		unfi 9102	. . . . 5 ⊢ (((𝐴 ∖ 𝐵) ∈ Fin ∧ 𝐵 ∈ Fin) → ((𝐴 ∖ 𝐵) ∪ 𝐵) ∈ Fin)
2		undif1 4411	. . . . . . 7 ⊢ ((𝐴 ∖ 𝐵) ∪ 𝐵) = (𝐴 ∪ 𝐵)
3	2	eleq1i 2831	. . . . . 6 ⊢ (((𝐴 ∖ 𝐵) ∪ 𝐵) ∈ Fin ↔ (𝐴 ∪ 𝐵) ∈ Fin)
4		unfir 9215	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → (𝐴 ∈ Fin ∧ 𝐵 ∈ Fin))
5	4	simpld 495	. . . . . 6 ⊢ ((𝐴 ∪ 𝐵) ∈ Fin → 𝐴 ∈ Fin)
6	3, 5	sylbi 218	. . . . 5 ⊢ (((𝐴 ∖ 𝐵) ∪ 𝐵) ∈ Fin → 𝐴 ∈ Fin)
7	1, 6	syl 17	. . . 4 ⊢ (((𝐴 ∖ 𝐵) ∈ Fin ∧ 𝐵 ∈ Fin) → 𝐴 ∈ Fin)
8	7	expcom 414	. . 3 ⊢ (𝐵 ∈ Fin → ((𝐴 ∖ 𝐵) ∈ Fin → 𝐴 ∈ Fin))
9	8	con3d 152	. 2 ⊢ (𝐵 ∈ Fin → (¬ 𝐴 ∈ Fin → ¬ (𝐴 ∖ 𝐵) ∈ Fin))
10	9	impcom 408	1 ⊢ ((¬ 𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → ¬ (𝐴 ∖ 𝐵) ∈ Fin)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2119 ∖ cdif 3887 ∪ cun 3888 Fincfn 8890

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pr 5369 ax-un 7685

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-om 7814 df-1o 8402 df-en 8891 df-fin 8894

This theorem is referenced by: ackbij1lem18 10156 bitsf1 16413 cusgrfilem3 29551 diffib 32616 hasheuni 34276 fineqvnttrclse 35312 topdifinffinlem 37716 eldioph2lem2 43217