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Theorem fin33i 9397

Description: Inference from isfin3-3 9396. (This is actually a bit stronger than isfin3-3 9396 because it does not assume 𝐹 is a set and does not use the Axiom of Infinity either.) (Contributed by Mario Carneiro, 17-May-2015.)

**Assertion**
Ref	Expression
fin33i	⊢ ((𝐴 ∈ Fin^III ∧ 𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) → ∩ ran 𝐹 ∈ ran 𝐹)

Distinct variable groups: 𝑥,𝐴 𝑥,𝐹

**Proof of Theorem fin33i**
Step	Hyp	Ref	Expression
1		isfin32i 9393	. . 3 ⊢ (𝐴 ∈ Fin^III → ¬ ω ≼^* 𝐴)
2	1	3ad2ant1 1127	. 2 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) → ¬ ω ≼^* 𝐴)
3		isf32lem11 9391	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ (𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹)) → ω ≼^* 𝐴)
4	3	3exp2 1447	. . 3 ⊢ (𝐴 ∈ Fin^III → (𝐹:ω⟶𝒫 𝐴 → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) → (¬ ∩ ran 𝐹 ∈ ran 𝐹 → ω ≼^* 𝐴))))
5	4	3imp 1101	. 2 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) → (¬ ∩ ran 𝐹 ∈ ran 𝐹 → ω ≼^* 𝐴))
6	2, 5	mt3d 142	1 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) → ∩ ran 𝐹 ∈ ran 𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1071 ∈ wcel 2145 ∀wral 3061 ⊆ wss 3723 𝒫 cpw 4298 ∩ cint 4612 class class class wbr 4787 ran crn 5251 suc csuc 5867 ⟶wf 6026 ‘cfv 6030 ωcom 7216 ≼^* cwdom 8622 Fin^IIIcfin3 9309

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-int 4613 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-tr 4888 df-id 5158 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-se 5210 df-we 5211 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-isom 6039 df-riota 6757 df-om 7217 df-wrecs 7563 df-recs 7625 df-rdg 7663 df-1o 7717 df-er 7900 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-wdom 8624 df-card 8969 df-fin4 9315 df-fin3 9316

This theorem is referenced by: isf34lem7 9407

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