Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wepwso | Structured version Visualization version GIF version |
Description: A well-ordering induces a strict ordering on the power set. EDITORIAL: when well-orderings are set like, this can be strengthened to remove 𝐴 ∈ 𝑉. (Contributed by Stefan O'Rear, 18-Jan-2015.) |
Ref | Expression |
---|---|
wepwso.t | ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦)))} |
Ref | Expression |
---|---|
wepwso | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2onn 7607 | . . . . 5 ⊢ 2𝑜 ∈ ω | |
2 | nnord 6965 | . . . . 5 ⊢ (2𝑜 ∈ ω → Ord 2𝑜) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ Ord 2𝑜 |
4 | ordwe 5653 | . . . 4 ⊢ (Ord 2𝑜 → E We 2𝑜) | |
5 | weso 5029 | . . . 4 ⊢ ( E We 2𝑜 → E Or 2𝑜) | |
6 | 3, 4, 5 | mp2b 10 | . . 3 ⊢ E Or 2𝑜 |
7 | eqid 2610 | . . . 4 ⊢ {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} | |
8 | 7 | wemapso 8339 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴 ∧ E Or 2𝑜) → {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} Or (2𝑜 ↑𝑚 𝐴)) |
9 | 6, 8 | mp3an3 1405 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} Or (2𝑜 ↑𝑚 𝐴)) |
10 | elex 3185 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
11 | wepwso.t | . . . . 5 ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑧 ∈ 𝑦 ∧ ¬ 𝑧 ∈ 𝑥) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑤 ∈ 𝑥 ↔ 𝑤 ∈ 𝑦)))} | |
12 | eqid 2610 | . . . . 5 ⊢ (𝑎 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑎 “ {1𝑜})) = (𝑎 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑎 “ {1𝑜})) | |
13 | 11, 7, 12 | wepwsolem 36630 | . . . 4 ⊢ (𝐴 ∈ V → (𝑎 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑎 “ {1𝑜})) Isom {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}, 𝑇((2𝑜 ↑𝑚 𝐴), 𝒫 𝐴)) |
14 | isoso 6498 | . . . 4 ⊢ ((𝑎 ∈ (2𝑜 ↑𝑚 𝐴) ↦ (◡𝑎 “ {1𝑜})) Isom {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))}, 𝑇((2𝑜 ↑𝑚 𝐴), 𝒫 𝐴) → ({〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} Or (2𝑜 ↑𝑚 𝐴) ↔ 𝑇 Or 𝒫 𝐴)) | |
15 | 10, 13, 14 | 3syl 18 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} Or (2𝑜 ↑𝑚 𝐴) ↔ 𝑇 Or 𝒫 𝐴)) |
16 | 15 | adantr 480 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → ({〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧) E (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} Or (2𝑜 ↑𝑚 𝐴) ↔ 𝑇 Or 𝒫 𝐴)) |
17 | 9, 16 | mpbid 221 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝑇 Or 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 {copab 4642 ↦ cmpt 4643 E cep 4947 Or wor 4958 We wwe 4996 ◡ccnv 5037 “ cima 5041 Ord word 5639 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 ωcom 6957 1𝑜c1o 7440 2𝑜c2o 7441 ↑𝑚 cmap 7744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-1o 7447 df-2o 7448 df-map 7746 |
This theorem is referenced by: aomclem1 36642 |
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