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Theorem fin33i 9780

Description: Inference from isfin3-3 9779. (This is actually a bit stronger than isfin3-3 9779 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 9776	. . 3 ⊢ (𝐴 ∈ Fin^III → ¬ ω ≼^* 𝐴)
2	1	3ad2ant1 1130	. 2 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) → ¬ ω ≼^* 𝐴)
3		isf32lem11 9774	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ (𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹)) → ω ≼^* 𝐴)
4	3	3exp2 1351	. . 3 ⊢ (𝐴 ∈ Fin^III → (𝐹:ω⟶𝒫 𝐴 → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) → (¬ ∩ ran 𝐹 ∈ ran 𝐹 → ω ≼^* 𝐴))))
5	4	3imp 1108	. 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 150	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 1084 ∈ wcel 2111 ∀wral 3106 ⊆ wss 3881 𝒫 cpw 4497 ∩ cint 4838 class class class wbr 5030 ran crn 5520 suc csuc 6161 ⟶wf 6320 ‘cfv 6324 ωcom 7560 ≼^* cwdom 9012 Fin^IIIcfin3 9692

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-wdom 9013 df-card 9352 df-fin4 9698 df-fin3 9699

This theorem is referenced by: isf34lem7 9790

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