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Theorem fin33i 10291

Description: Inference from isfin3-3 10290. (This is actually a bit stronger than isfin3-3 10290 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 10287	. . 3 ⊢ (𝐴 ∈ Fin^III → ¬ ω ≼^* 𝐴)
2	1	3ad2ant1 1134	. 2 ⊢ ((𝐴 ∈ Fin^III ∧ 𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥)) → ¬ ω ≼^* 𝐴)
3		isf32lem11 10285	. . . 4 ⊢ ((𝐴 ∈ Fin^III ∧ (𝐹:ω⟶𝒫 𝐴 ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) ∧ ¬ ∩ ran 𝐹 ∈ ran 𝐹)) → ω ≼^* 𝐴)
4	3	3exp2 1356	. . 3 ⊢ (𝐴 ∈ Fin^III → (𝐹:ω⟶𝒫 𝐴 → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) → (¬ ∩ ran 𝐹 ∈ ran 𝐹 → ω ≼^* 𝐴))))
5	4	3imp 1111	. 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 148	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 1087 ∈ wcel 2114 ∀wral 3052 ⊆ wss 3903 𝒫 cpw 4556 ∩ cint 4904 class class class wbr 5100 ran crn 5633 suc csuc 6327 ⟶wf 6496 ‘cfv 6500 ωcom 7818 ≼^* cwdom 9481 Fin^IIIcfin3 10203

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-wdom 9482 df-card 9863 df-fin4 10209 df-fin3 10210

This theorem is referenced by: isf34lem7 10301

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