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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 10499. See dominf 10485 for a version proved from ax-cc 10475. (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 3003 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≠ ∅ ↔ 𝐴 ≠ ∅)) | |
| 3 | id 22 | . . . . 5 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
| 4 | unieq 4918 | . . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴) | |
| 5 | 3, 4 | sseq12d 4017 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ ∪ 𝑥 ↔ 𝐴 ⊆ ∪ 𝐴)) |
| 6 | 2, 5 | anbi12d 632 | . . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ (𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴))) |
| 7 | breq2 5147 | . . 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 2737 | . . . 4 ⊢ (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) = (𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}) | |
| 10 | eqid 2737 | . . . 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 9673 | . . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) → (rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥) |
| 12 | vpwex 5377 | . . . 4 ⊢ 𝒫 𝑥 ∈ V | |
| 13 | 12 | f1dom 9014 | . . 3 ⊢ ((rec((𝑦 ∈ V ↦ {𝑤 ∈ 𝑥 ∣ (𝑤 ∩ 𝑥) ⊆ 𝑦}), ∅) ↾ ω):ω–1-1→𝒫 𝑥 → ω ≼ 𝒫 𝑥) |
| 14 | pwfi 9357 | . . . . . . 7 ⊢ (𝑥 ∈ Fin ↔ 𝒫 𝑥 ∈ Fin) | |
| 15 | 14 | biimpi 216 | . . . . . 6 ⊢ (𝑥 ∈ Fin → 𝒫 𝑥 ∈ Fin) |
| 16 | isfinite 9692 | . . . . . 6 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω) | |
| 17 | isfinite 9692 | . . . . . 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 9683 | . . . . 5 ⊢ ω ∈ V | |
| 21 | domtri 10596 | . . . . 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 3484 | . . . . 5 ⊢ 𝑥 ∈ V | |
| 24 | domtri 10596 | . . . . 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 3558 | 1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ ∪ 𝐴) → ω ≼ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 {crab 3436 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 ↾ cres 5687 –1-1→wf1 6558 ωcom 7887 reccrdg 8449 ≼ cdom 8983 ≺ csdm 8984 Fincfn 8985 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-reg 9632 ax-inf2 9681 ax-ac2 10503 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-ac 10156 |
| This theorem is referenced by: (None) |
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