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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 9473. See dominf 9459 for a version proved from ax-cc 9449. (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 2994 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≠ ∅ ↔ 𝐴 ≠ ∅)) | |
3 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
4 | unieq 4596 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
5 | 3, 4 | sseq12d 3775 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ 𝑥 ↔ 𝐴 ⊆ ∪ 𝐴)) |
6 | 2, 5 | anbi12d 749 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ (𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴))) |
7 | breq2 4808 | . . 3 ⊢ (𝑥 = 𝐴 → (ω ≼ 𝑥 ↔ ω ≼ 𝐴)) | |
8 | 6, 7 | imbi12d 333 | . 2 ⊢ (𝑥 = 𝐴 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ≼ 𝑥) ↔ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴))) |
9 | eqid 2760 | . . . 4 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) | |
10 | eqid 2760 | . . . 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 8703 | . . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥) |
12 | vpwex 4998 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
13 | 12 | f1dom 8143 | . . 3 ⊢ ((rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥 → ω ≼ 𝒫 𝑥) |
14 | pwfi 8426 | . . . . . . 7 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin) | |
15 | 14 | biimpi 206 | . . . . . 6 ⊢ (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin) |
16 | isfinite 8722 | . . . . . 6 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω) | |
17 | isfinite 8722 | . . . . . 6 ⊢ (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑥 ≺ ω) | |
18 | 15, 16, 17 | 3imtr3i 280 | . . . . 5 ⊢ (𝑥 ≺ ω → 𝒫 𝑥 ≺ ω) |
19 | 18 | con3i 150 | . . . 4 ⊢ (¬ 𝒫 𝑥 ≺ ω → ¬ 𝑥 ≺ ω) |
20 | omex 8713 | . . . . 5 ⊢ ω ∈ V | |
21 | domtri 9570 | . . . . 5 ⊢ ((ω ∈ V ∧ 𝒫 𝑥 ∈ V) → (ω ≼ 𝒫 𝑥 ↔ ¬ 𝒫 𝑥 ≺ ω)) | |
22 | 20, 12, 21 | mp2an 710 | . . . 4 ⊢ (ω ≼ 𝒫 𝑥 ↔ ¬ 𝒫 𝑥 ≺ ω) |
23 | vex 3343 | . . . . 5 ⊢ 𝑥 ∈ V | |
24 | domtri 9570 | . . . . 5 ⊢ ((ω ∈ V ∧ 𝑥 ∈ V) → (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω)) | |
25 | 20, 23, 24 | mp2an 710 | . . . 4 ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω) |
26 | 19, 22, 25 | 3imtr4i 281 | . . 3 ⊢ (ω ≼ 𝒫 𝑥 → ω ≼ 𝑥) |
27 | 11, 13, 26 | 3syl 18 | . 2 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ≼ 𝑥) |
28 | 1, 8, 27 | vtocl 3399 | 1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 {crab 3054 Vcvv 3340 ∩ cin 3714 ⊆ wss 3715 ∅c0 4058 𝒫 cpw 4302 ∪ cuni 4588 class class class wbr 4804 ↦ cmpt 4881 ↾ cres 5268 –1-1→wf1 6046 ωcom 7230 reccrdg 7674 ≼ cdom 8119 ≺ csdm 8120 Fincfn 8121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-reg 8662 ax-inf2 8711 ax-ac2 9477 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-card 8955 df-ac 9129 |
This theorem is referenced by: (None) |
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