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Mirrors > Home > MPE Home > Th. List > inawinalem | Structured version Visualization version GIF version |
Description: Lemma for inawina 10101. (Contributed by Mario Carneiro, 8-Jun-2014.) |
Ref | Expression |
---|---|
inawinalem | ⊢ (𝐴 ∈ On → (∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomdom 8520 | . . . . 5 ⊢ (𝒫 𝑥 ≺ 𝐴 → 𝒫 𝑥 ≼ 𝐴) | |
2 | ondomen 9448 | . . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝒫 𝑥 ≼ 𝐴) → 𝒫 𝑥 ∈ dom card) | |
3 | isnum2 9358 | . . . . . 6 ⊢ (𝒫 𝑥 ∈ dom card ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝒫 𝑥) | |
4 | 2, 3 | sylib 221 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝒫 𝑥 ≼ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝒫 𝑥) |
5 | 1, 4 | sylan2 595 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝒫 𝑥 ≺ 𝐴) → ∃𝑦 ∈ On 𝑦 ≈ 𝒫 𝑥) |
6 | ensdomtr 8637 | . . . . . . . . 9 ⊢ ((𝑦 ≈ 𝒫 𝑥 ∧ 𝒫 𝑥 ≺ 𝐴) → 𝑦 ≺ 𝐴) | |
7 | 6 | ad2ant2l 745 | . . . . . . . 8 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≈ 𝒫 𝑥) ∧ (𝐴 ∈ On ∧ 𝒫 𝑥 ≺ 𝐴)) → 𝑦 ≺ 𝐴) |
8 | sdomel 8648 | . . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ≺ 𝐴 → 𝑦 ∈ 𝐴)) | |
9 | 8 | ad2ant2r 746 | . . . . . . . 8 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≈ 𝒫 𝑥) ∧ (𝐴 ∈ On ∧ 𝒫 𝑥 ≺ 𝐴)) → (𝑦 ≺ 𝐴 → 𝑦 ∈ 𝐴)) |
10 | 7, 9 | mpd 15 | . . . . . . 7 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≈ 𝒫 𝑥) ∧ (𝐴 ∈ On ∧ 𝒫 𝑥 ≺ 𝐴)) → 𝑦 ∈ 𝐴) |
11 | vex 3444 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
12 | 11 | canth2 8654 | . . . . . . . . 9 ⊢ 𝑥 ≺ 𝒫 𝑥 |
13 | ensym 8541 | . . . . . . . . 9 ⊢ (𝑦 ≈ 𝒫 𝑥 → 𝒫 𝑥 ≈ 𝑦) | |
14 | sdomentr 8635 | . . . . . . . . 9 ⊢ ((𝑥 ≺ 𝒫 𝑥 ∧ 𝒫 𝑥 ≈ 𝑦) → 𝑥 ≺ 𝑦) | |
15 | 12, 13, 14 | sylancr 590 | . . . . . . . 8 ⊢ (𝑦 ≈ 𝒫 𝑥 → 𝑥 ≺ 𝑦) |
16 | 15 | ad2antlr 726 | . . . . . . 7 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≈ 𝒫 𝑥) ∧ (𝐴 ∈ On ∧ 𝒫 𝑥 ≺ 𝐴)) → 𝑥 ≺ 𝑦) |
17 | 10, 16 | jca 515 | . . . . . 6 ⊢ (((𝑦 ∈ On ∧ 𝑦 ≈ 𝒫 𝑥) ∧ (𝐴 ∈ On ∧ 𝒫 𝑥 ≺ 𝐴)) → (𝑦 ∈ 𝐴 ∧ 𝑥 ≺ 𝑦)) |
18 | 17 | expcom 417 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝒫 𝑥 ≺ 𝐴) → ((𝑦 ∈ On ∧ 𝑦 ≈ 𝒫 𝑥) → (𝑦 ∈ 𝐴 ∧ 𝑥 ≺ 𝑦))) |
19 | 18 | reximdv2 3230 | . . . 4 ⊢ ((𝐴 ∈ On ∧ 𝒫 𝑥 ≺ 𝐴) → (∃𝑦 ∈ On 𝑦 ≈ 𝒫 𝑥 → ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) |
20 | 5, 19 | mpd 15 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝒫 𝑥 ≺ 𝐴) → ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦) |
21 | 20 | ex 416 | . 2 ⊢ (𝐴 ∈ On → (𝒫 𝑥 ≺ 𝐴 → ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) |
22 | 21 | ralimdv 3145 | 1 ⊢ (𝐴 ∈ On → (∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐴 𝑥 ≺ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 𝒫 cpw 4497 class class class wbr 5030 dom cdm 5519 Oncon0 6159 ≈ cen 8489 ≼ cdom 8490 ≺ csdm 8491 cardccrd 9348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-wrecs 7930 df-recs 7991 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-card 9352 |
This theorem is referenced by: inawina 10101 tskcard 10192 gruina 10229 |
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