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Mirrors > Home > MPE Home > Th. List > elharval | Structured version Visualization version GIF version |
Description: The Hartogs number of a set contains exactly the ordinals that set dominates. Combined with harcl 9153, this implies that the Hartogs number of a set is greater than all ordinals that set dominates. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 15-May-2015.) |
Ref | Expression |
---|---|
elharval | ⊢ (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6728 | . 2 ⊢ (𝑌 ∈ (har‘𝑋) → 𝑋 ∈ V) | |
2 | reldom 8610 | . . . 4 ⊢ Rel ≼ | |
3 | 2 | brrelex2i 5591 | . . 3 ⊢ (𝑌 ≼ 𝑋 → 𝑋 ∈ V) |
4 | 3 | adantl 485 | . 2 ⊢ ((𝑌 ∈ On ∧ 𝑌 ≼ 𝑋) → 𝑋 ∈ V) |
5 | harval 9154 | . . . 4 ⊢ (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋}) | |
6 | 5 | eleq2d 2816 | . . 3 ⊢ (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ 𝑌 ∈ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋})) |
7 | breq1 5042 | . . . 4 ⊢ (𝑦 = 𝑌 → (𝑦 ≼ 𝑋 ↔ 𝑌 ≼ 𝑋)) | |
8 | 7 | elrab 3591 | . . 3 ⊢ (𝑌 ∈ {𝑦 ∈ On ∣ 𝑦 ≼ 𝑋} ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋)) |
9 | 6, 8 | bitrdi 290 | . 2 ⊢ (𝑋 ∈ V → (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋))) |
10 | 1, 4, 9 | pm5.21nii 383 | 1 ⊢ (𝑌 ∈ (har‘𝑋) ↔ (𝑌 ∈ On ∧ 𝑌 ≼ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∈ wcel 2112 {crab 3055 Vcvv 3398 class class class wbr 5039 Oncon0 6191 ‘cfv 6358 ≼ cdom 8602 harchar 9150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-wrecs 8025 df-recs 8086 df-en 8605 df-dom 8606 df-oi 9104 df-har 9151 |
This theorem is referenced by: harndom 9156 harcard 9559 cardprclem 9560 cardaleph 9668 dfac12lem2 9723 hsmexlem1 10005 pwcfsdom 10162 pwfseqlem5 10242 hargch 10252 harinf 40500 harn0 40571 |
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