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Mirrors > Home > MPE Home > Th. List > dfac8b | Structured version Visualization version GIF version |
Description: The well-ordering theorem: every numerable set is well-orderable. (Contributed by Mario Carneiro, 5-Jan-2013.) (Revised by Mario Carneiro, 29-Apr-2015.) |
Ref | Expression |
---|---|
dfac8b | ⊢ (𝐴 ∈ dom card → ∃𝑥 𝑥 We 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cardid2 9991 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
2 | bren 8994 | . . 3 ⊢ ((card‘𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴) | |
3 | 1, 2 | sylib 218 | . 2 ⊢ (𝐴 ∈ dom card → ∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴) |
4 | sqxpexg 7774 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V) | |
5 | incom 4217 | . . . . . 6 ⊢ ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}) | |
6 | inex1g 5325 | . . . . . 6 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}) ∈ V) | |
7 | 5, 6 | eqeltrid 2843 | . . . . 5 ⊢ ((𝐴 × 𝐴) ∈ V → ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) ∈ V) |
8 | 4, 7 | syl 17 | . . . 4 ⊢ (𝐴 ∈ dom card → ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) ∈ V) |
9 | f1ocnv 6861 | . . . . . 6 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→(card‘𝐴)) | |
10 | cardon 9982 | . . . . . . . 8 ⊢ (card‘𝐴) ∈ On | |
11 | 10 | onordi 6497 | . . . . . . 7 ⊢ Ord (card‘𝐴) |
12 | ordwe 6399 | . . . . . . 7 ⊢ (Ord (card‘𝐴) → E We (card‘𝐴)) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ E We (card‘𝐴) |
14 | eqid 2735 | . . . . . . 7 ⊢ {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} = {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} | |
15 | 14 | f1owe 7373 | . . . . . 6 ⊢ (◡𝑓:𝐴–1-1-onto→(card‘𝐴) → ( E We (card‘𝐴) → {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴)) |
16 | 9, 13, 15 | mpisyl 21 | . . . . 5 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴) |
17 | weinxp 5773 | . . . . 5 ⊢ ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} We 𝐴 ↔ ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴) | |
18 | 16, 17 | sylib 218 | . . . 4 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴) |
19 | weeq1 5676 | . . . . 5 ⊢ (𝑥 = ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) → (𝑥 We 𝐴 ↔ ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴)) | |
20 | 19 | spcegv 3597 | . . . 4 ⊢ (({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) ∈ V → (({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴 → ∃𝑥 𝑥 We 𝐴)) |
21 | 8, 18, 20 | syl2im 40 | . . 3 ⊢ (𝐴 ∈ dom card → (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ∃𝑥 𝑥 We 𝐴)) |
22 | 21 | exlimdv 1931 | . 2 ⊢ (𝐴 ∈ dom card → (∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴 → ∃𝑥 𝑥 We 𝐴)) |
23 | 3, 22 | mpd 15 | 1 ⊢ (𝐴 ∈ dom card → ∃𝑥 𝑥 We 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 class class class wbr 5148 {copab 5210 E cep 5588 We wwe 5640 × cxp 5687 ◡ccnv 5688 dom cdm 5689 Ord word 6385 –1-1-onto→wf1o 6562 ‘cfv 6563 ≈ cen 8981 cardccrd 9973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-en 8985 df-card 9977 |
This theorem is referenced by: ween 10073 ac5num 10074 dfac8 10174 numiunnum 36453 |
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