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Mirrors > Home > MPE Home > Th. List > fin12 | Structured version Visualization version GIF version |
Description: Weak theorem which skips Ia but has a trivial proof, needed to prove fin1a2 9836. (Contributed by Stefan O'Rear, 8-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
fin12 | ⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinII) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3497 | . . . . . . . 8 ⊢ 𝑏 ∈ V | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝑏 ∈ 𝒫 𝒫 𝐴) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → 𝑏 ∈ V) |
3 | isfin1-3 9807 | . . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (𝐴 ∈ Fin ↔ ◡ [⊊] Fr 𝒫 𝐴)) | |
4 | 3 | ibi 269 | . . . . . . . 8 ⊢ (𝐴 ∈ Fin → ◡ [⊊] Fr 𝒫 𝐴) |
5 | 4 | ad2antrr 724 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝑏 ∈ 𝒫 𝒫 𝐴) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → ◡ [⊊] Fr 𝒫 𝐴) |
6 | elpwi 4547 | . . . . . . . 8 ⊢ (𝑏 ∈ 𝒫 𝒫 𝐴 → 𝑏 ⊆ 𝒫 𝐴) | |
7 | 6 | ad2antlr 725 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝑏 ∈ 𝒫 𝒫 𝐴) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → 𝑏 ⊆ 𝒫 𝐴) |
8 | simprl 769 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝑏 ∈ 𝒫 𝒫 𝐴) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → 𝑏 ≠ ∅) | |
9 | fri 5516 | . . . . . . 7 ⊢ (((𝑏 ∈ V ∧ ◡ [⊊] Fr 𝒫 𝐴) ∧ (𝑏 ⊆ 𝒫 𝐴 ∧ 𝑏 ≠ ∅)) → ∃𝑐 ∈ 𝑏 ∀𝑑 ∈ 𝑏 ¬ 𝑑◡ [⊊] 𝑐) | |
10 | 2, 5, 7, 8, 9 | syl22anc 836 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝑏 ∈ 𝒫 𝒫 𝐴) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → ∃𝑐 ∈ 𝑏 ∀𝑑 ∈ 𝑏 ¬ 𝑑◡ [⊊] 𝑐) |
11 | vex 3497 | . . . . . . . . . . 11 ⊢ 𝑑 ∈ V | |
12 | vex 3497 | . . . . . . . . . . 11 ⊢ 𝑐 ∈ V | |
13 | 11, 12 | brcnv 5752 | . . . . . . . . . 10 ⊢ (𝑑◡ [⊊] 𝑐 ↔ 𝑐 [⊊] 𝑑) |
14 | 11 | brrpss 7451 | . . . . . . . . . 10 ⊢ (𝑐 [⊊] 𝑑 ↔ 𝑐 ⊊ 𝑑) |
15 | 13, 14 | bitri 277 | . . . . . . . . 9 ⊢ (𝑑◡ [⊊] 𝑐 ↔ 𝑐 ⊊ 𝑑) |
16 | 15 | notbii 322 | . . . . . . . 8 ⊢ (¬ 𝑑◡ [⊊] 𝑐 ↔ ¬ 𝑐 ⊊ 𝑑) |
17 | 16 | ralbii 3165 | . . . . . . 7 ⊢ (∀𝑑 ∈ 𝑏 ¬ 𝑑◡ [⊊] 𝑐 ↔ ∀𝑑 ∈ 𝑏 ¬ 𝑐 ⊊ 𝑑) |
18 | 17 | rexbii 3247 | . . . . . 6 ⊢ (∃𝑐 ∈ 𝑏 ∀𝑑 ∈ 𝑏 ¬ 𝑑◡ [⊊] 𝑐 ↔ ∃𝑐 ∈ 𝑏 ∀𝑑 ∈ 𝑏 ¬ 𝑐 ⊊ 𝑑) |
19 | 10, 18 | sylib 220 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝑏 ∈ 𝒫 𝒫 𝐴) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → ∃𝑐 ∈ 𝑏 ∀𝑑 ∈ 𝑏 ¬ 𝑐 ⊊ 𝑑) |
20 | sorpssuni 7457 | . . . . . 6 ⊢ ( [⊊] Or 𝑏 → (∃𝑐 ∈ 𝑏 ∀𝑑 ∈ 𝑏 ¬ 𝑐 ⊊ 𝑑 ↔ ∪ 𝑏 ∈ 𝑏)) | |
21 | 20 | ad2antll 727 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝑏 ∈ 𝒫 𝒫 𝐴) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → (∃𝑐 ∈ 𝑏 ∀𝑑 ∈ 𝑏 ¬ 𝑐 ⊊ 𝑑 ↔ ∪ 𝑏 ∈ 𝑏)) |
22 | 19, 21 | mpbid 234 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝑏 ∈ 𝒫 𝒫 𝐴) ∧ (𝑏 ≠ ∅ ∧ [⊊] Or 𝑏)) → ∪ 𝑏 ∈ 𝑏) |
23 | 22 | ex 415 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝑏 ∈ 𝒫 𝒫 𝐴) → ((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏)) |
24 | 23 | ralrimiva 3182 | . 2 ⊢ (𝐴 ∈ Fin → ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏)) |
25 | isfin2 9715 | . 2 ⊢ (𝐴 ∈ Fin → (𝐴 ∈ FinII ↔ ∀𝑏 ∈ 𝒫 𝒫 𝐴((𝑏 ≠ ∅ ∧ [⊊] Or 𝑏) → ∪ 𝑏 ∈ 𝑏))) | |
26 | 24, 25 | mpbird 259 | 1 ⊢ (𝐴 ∈ Fin → 𝐴 ∈ FinII) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⊆ wss 3935 ⊊ wpss 3936 ∅c0 4290 𝒫 cpw 4538 ∪ cuni 4837 class class class wbr 5065 Or wor 5472 Fr wfr 5510 ◡ccnv 5553 [⊊] crpss 7447 Fincfn 8508 FinIIcfin2 9700 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-rpss 7448 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-er 8288 df-map 8407 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fin2 9707 |
This theorem is referenced by: fin1a2s 9835 fin1a2 9836 finngch 10076 |
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