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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 9382 | . . 3 ⊢ (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴) | |
2 | bren 8518 | . . 3 ⊢ ((card‘𝐴) ≈ 𝐴 ↔ ∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴) | |
3 | 1, 2 | sylib 220 | . 2 ⊢ (𝐴 ∈ dom card → ∃𝑓 𝑓:(card‘𝐴)–1-1-onto→𝐴) |
4 | sqxpexg 7477 | . . . . 5 ⊢ (𝐴 ∈ dom card → (𝐴 × 𝐴) ∈ V) | |
5 | incom 4178 | . . . . . 6 ⊢ ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) = ((𝐴 × 𝐴) ∩ {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}) | |
6 | inex1g 5223 | . . . . . 6 ⊢ ((𝐴 × 𝐴) ∈ V → ((𝐴 × 𝐴) ∩ {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)}) ∈ V) | |
7 | 5, 6 | eqeltrid 2917 | . . . . 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 6627 | . . . . . 6 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ◡𝑓:𝐴–1-1-onto→(card‘𝐴)) | |
10 | cardon 9373 | . . . . . . . 8 ⊢ (card‘𝐴) ∈ On | |
11 | 10 | onordi 6295 | . . . . . . 7 ⊢ Ord (card‘𝐴) |
12 | ordwe 6204 | . . . . . . 7 ⊢ (Ord (card‘𝐴) → E We (card‘𝐴)) | |
13 | 11, 12 | ax-mp 5 | . . . . . 6 ⊢ E We (card‘𝐴) |
14 | eqid 2821 | . . . . . . 7 ⊢ {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} = {〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} | |
15 | 14 | f1owe 7106 | . . . . . 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 5636 | . . . . 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 220 | . . . 4 ⊢ (𝑓:(card‘𝐴)–1-1-onto→𝐴 → ({〈𝑧, 𝑤〉 ∣ (◡𝑓‘𝑧) E (◡𝑓‘𝑤)} ∩ (𝐴 × 𝐴)) We 𝐴) |
19 | weeq1 5543 | . . . . 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 1934 | . 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 1780 ∈ wcel 2114 Vcvv 3494 ∩ cin 3935 class class class wbr 5066 {copab 5128 E cep 5464 We wwe 5513 × cxp 5553 ◡ccnv 5554 dom cdm 5555 Ord word 6190 –1-1-onto→wf1o 6354 ‘cfv 6355 ≈ cen 8506 cardccrd 9364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-ord 6194 df-on 6195 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-en 8510 df-card 9368 |
This theorem is referenced by: ween 9461 ac5num 9462 dfac8 9561 |
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