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Mirrors > Home > MPE Home > Th. List > dominf | 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-cc 9592. See dominfac 9730 for a version proved from ax-ac 9616. The axiom of Regularity is used for this proof, via inf3lem6 8827, and its use is necessary: otherwise the set 𝐴 = {𝐴} or 𝐴 = {∅, 𝐴} (where the second example even has nonempty well-founded part) provides a counterexample. (Contributed by Mario Carneiro, 9-Feb-2013.) |
Ref | Expression |
---|---|
dominf.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
dominf | ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dominf.1 | . 2 ⊢ 𝐴 ∈ V | |
2 | neeq1 3031 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≠ ∅ ↔ 𝐴 ≠ ∅)) | |
3 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
4 | unieq 4679 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
5 | 3, 4 | sseq12d 3853 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ 𝑥 ↔ 𝐴 ⊆ ∪ 𝐴)) |
6 | 2, 5 | anbi12d 624 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ (𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴))) |
7 | breq2 4890 | . . 3 ⊢ (𝑥 = 𝐴 → (ω ≼ 𝑥 ↔ ω ≼ 𝐴)) | |
8 | 6, 7 | imbi12d 336 | . 2 ⊢ (𝑥 = 𝐴 → (((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ≼ 𝑥) ↔ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴))) |
9 | eqid 2778 | . . . 4 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) | |
10 | eqid 2778 | . . . 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 8827 | . . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥) |
12 | vpwex 5089 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
13 | 12 | f1dom 8263 | . . 3 ⊢ ((rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥 → ω ≼ 𝒫 𝑥) |
14 | pwfi 8549 | . . . . . . 7 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin) | |
15 | 14 | biimpi 208 | . . . . . 6 ⊢ (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin) |
16 | isfinite 8846 | . . . . . 6 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω) | |
17 | isfinite 8846 | . . . . . 6 ⊢ (𝒫 𝑥 ∈ Fin ↔ 𝒫 𝑥 ≺ ω) | |
18 | 15, 16, 17 | 3imtr3i 283 | . . . . 5 ⊢ (𝑥 ≺ ω → 𝒫 𝑥 ≺ ω) |
19 | 18 | con3i 152 | . . . 4 ⊢ (¬ 𝒫 𝑥 ≺ ω → ¬ 𝑥 ≺ ω) |
20 | 12 | domtriom 9600 | . . . 4 ⊢ (ω ≼ 𝒫 𝑥 ↔ ¬ 𝒫 𝑥 ≺ ω) |
21 | vex 3401 | . . . . 5 ⊢ 𝑥 ∈ V | |
22 | 21 | domtriom 9600 | . . . 4 ⊢ (ω ≼ 𝑥 ↔ ¬ 𝑥 ≺ ω) |
23 | 19, 20, 22 | 3imtr4i 284 | . . 3 ⊢ (ω ≼ 𝒫 𝑥 → ω ≼ 𝑥) |
24 | 11, 13, 23 | 3syl 18 | . 2 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → ω ≼ 𝑥) |
25 | 1, 8, 24 | vtocl 3460 | 1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 {crab 3094 Vcvv 3398 ∩ cin 3791 ⊆ wss 3792 ∅c0 4141 𝒫 cpw 4379 ∪ cuni 4671 class class class wbr 4886 ↦ cmpt 4965 ↾ cres 5357 –1-1→wf1 6132 ωcom 7343 reccrdg 7788 ≼ cdom 8239 ≺ csdm 8240 Fincfn 8241 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-reg 8786 ax-inf2 8835 ax-cc 9592 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-cda 9325 |
This theorem is referenced by: axgroth3 9988 |
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