Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dominfac | Structured version Visualization version GIF version |
Description: A nonempty set that is a subset of its union is infinite. This version is proved from ax-ac 9880. See dominf 9866 for a version proved from ax-cc 9856. (Contributed by NM, 25-Mar-2007.) |
Ref | Expression |
---|---|
dominfac.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
dominfac | ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dominfac.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | neeq1 3078 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≠ ∅ ↔ 𝐴 ≠ ∅)) | |
3 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
4 | unieq 4848 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
5 | 3, 4 | sseq12d 3999 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ 𝑥 ↔ 𝐴 ⊆ ∪ 𝐴)) |
6 | 2, 5 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ (𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴))) |
7 | breq2 5069 | . . 3 ⊢ (𝑥 = 𝐴 → (ω ≼ 𝑥 ↔ ω ≼ 𝐴)) | |
8 | 6, 7 | imbi12d 347 | . 2 ⊢ (𝑥 = 𝐴 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ≼ 𝑥) ↔ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴))) |
9 | eqid 2821 | . . . 4 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) | |
10 | eqid 2821 | . . . 4 ⊢ (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω) = (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω) | |
11 | 9, 10, 1, 1 | inf3lem6 9095 | . . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥) |
12 | vpwex 5277 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
13 | 12 | f1dom 8530 | . . 3 ⊢ ((rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥 → ω ≼ 𝒫 𝑥) |
14 | pwfi 8818 | . . . . . . 7 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin) | |
15 | 14 | biimpi 218 | . . . . . 6 ⊢ (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin) |
16 | isfinite 9114 | . . . . . 6 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω) | |
17 | isfinite 9114 | . . . . . 6 ⊢ (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑥 ≺ ω) | |
18 | 15, 16, 17 | 3imtr3i 293 | . . . . 5 ⊢ (𝑥 ≺ ω → 𝒫 𝑥 ≺ ω) |
19 | 18 | con3i 157 | . . . 4 ⊢ (¬ 𝒫 𝑥 ≺ ω → ¬ 𝑥 ≺ ω) |
20 | omex 9105 | . . . . 5 ⊢ ω ∈ V | |
21 | domtri 9977 | . . . . 5 ⊢ ((ω ∈ V ∧ 𝒫 𝑥 ∈ V) → (ω ≼ 𝒫 𝑥 ↔ ¬ 𝒫 𝑥 ≺ ω)) | |
22 | 20, 12, 21 | mp2an 690 | . . . 4 ⊢ (ω ≼ 𝒫 𝑥 ↔ ¬ 𝒫 𝑥 ≺ ω) |
23 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
24 | domtri 9977 | . . . . 5 ⊢ ((ω ∈ V ∧ 𝑥 ∈ V) → (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)) | |
25 | 20, 23, 24 | mp2an 690 | . . . 4 ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω) |
26 | 19, 22, 25 | 3imtr4i 294 | . . 3 ⊢ (ω ≼ 𝒫 𝑥 → ω ≼ 𝑥) |
27 | 11, 13, 26 | 3syl 18 | . 2 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ≼ 𝑥) |
28 | 1, 8, 27 | vtocl 3559 | 1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 {crab 3142 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 ∪ cuni 4837 class class class wbr 5065 ↦ cmpt 5145 ↾ cres 5556 –1-1→wf1 6351 ωcom 7579 reccrdg 8044 ≼ cdom 8506 ≺ csdm 8507 Fincfn 8508 |
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-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-reg 9055 ax-inf2 9103 ax-ac2 9884 |
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-rmo 3146 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-se 5514 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-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 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-card 9367 df-ac 9541 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |