![]() |
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 10496. See dominf 10482 for a version proved from ax-cc 10472. (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 3000 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≠ ∅ ↔ 𝐴 ≠ ∅)) | |
3 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
4 | unieq 4922 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
5 | 3, 4 | sseq12d 4028 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ 𝑥 ↔ 𝐴 ⊆ ∪ 𝐴)) |
6 | 2, 5 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ (𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴))) |
7 | breq2 5151 | . . 3 ⊢ (𝑥 = 𝐴 → (ω ≼ 𝑥 ↔ ω ≼ 𝐴)) | |
8 | 6, 7 | imbi12d 344 | . 2 ⊢ (𝑥 = 𝐴 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ≼ 𝑥) ↔ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴))) |
9 | eqid 2734 | . . . 4 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) | |
10 | eqid 2734 | . . . 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 9670 | . . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥) |
12 | vpwex 5382 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
13 | 12 | f1dom 9012 | . . 3 ⊢ ((rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥 → ω ≼ 𝒫 𝑥) |
14 | pwfi 9354 | . . . . . . 7 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin) | |
15 | 14 | biimpi 216 | . . . . . 6 ⊢ (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin) |
16 | isfinite 9689 | . . . . . 6 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω) | |
17 | isfinite 9689 | . . . . . 6 ⊢ (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑥 ≺ ω) | |
18 | 15, 16, 17 | 3imtr3i 291 | . . . . 5 ⊢ (𝑥 ≺ ω → 𝒫 𝑥 ≺ ω) |
19 | 18 | con3i 154 | . . . 4 ⊢ (¬ 𝒫 𝑥 ≺ ω → ¬ 𝑥 ≺ ω) |
20 | omex 9680 | . . . . 5 ⊢ ω ∈ V | |
21 | domtri 10593 | . . . . 5 ⊢ ((ω ∈ V ∧ 𝒫 𝑥 ∈ V) → (ω ≼ 𝒫 𝑥 ↔ ¬ 𝒫 𝑥 ≺ ω)) | |
22 | 20, 12, 21 | mp2an 692 | . . . 4 ⊢ (ω ≼ 𝒫 𝑥 ↔ ¬ 𝒫 𝑥 ≺ ω) |
23 | vex 3481 | . . . . 5 ⊢ 𝑥 ∈ V | |
24 | domtri 10593 | . . . . 5 ⊢ ((ω ∈ V ∧ 𝑥 ∈ V) → (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)) | |
25 | 20, 23, 24 | mp2an 692 | . . . 4 ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω) |
26 | 19, 22, 25 | 3imtr4i 292 | . . 3 ⊢ (ω ≼ 𝒫 𝑥 → ω ≼ 𝑥) |
27 | 11, 13, 26 | 3syl 18 | . 2 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ≼ 𝑥) |
28 | 1, 8, 27 | vtocl 3557 | 1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 {crab 3432 Vcvv 3477 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 𝒫 cpw 4604 ∪ cuni 4911 class class class wbr 5147 ↦ cmpt 5230 ↾ cres 5690 –1-1→wf1 6559 ωcom 7886 reccrdg 8447 ≼ cdom 8981 ≺ csdm 8982 Fincfn 8983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-reg 9629 ax-inf2 9678 ax-ac2 10500 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-card 9976 df-ac 10153 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |