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Mirrors > Home > MPE Home > Th. List > r1elwf | Structured version Visualization version GIF version |
Description: Any member of the cumulative hierarchy is well-founded. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
r1elwf | ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1funlim 8512 | . . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
2 | 1 | simpri 477 | . . . . 5 ⊢ Lim dom 𝑅1 |
3 | limord 5701 | . . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
4 | ordsson 6881 | . . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On) | |
5 | 2, 3, 4 | mp2b 10 | . . . 4 ⊢ dom 𝑅1 ⊆ On |
6 | elfvdm 6130 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1) | |
7 | 5, 6 | sseldi 3566 | . . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ On) |
8 | r1tr 8522 | . . . . . 6 ⊢ Tr (𝑅1‘𝐵) | |
9 | trss 4689 | . . . . . 6 ⊢ (Tr (𝑅1‘𝐵) → (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ⊆ (𝑅1‘𝐵))) | |
10 | 8, 9 | ax-mp 5 | . . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ⊆ (𝑅1‘𝐵)) |
11 | elpwg 4116 | . . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝐴 ∈ 𝒫 (𝑅1‘𝐵) ↔ 𝐴 ⊆ (𝑅1‘𝐵))) | |
12 | 10, 11 | mpbird 246 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ 𝒫 (𝑅1‘𝐵)) |
13 | r1sucg 8515 | . . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → (𝑅1‘suc 𝐵) = 𝒫 (𝑅1‘𝐵)) | |
14 | 6, 13 | syl 17 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝑅1‘suc 𝐵) = 𝒫 (𝑅1‘𝐵)) |
15 | 12, 14 | eleqtrrd 2691 | . . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ (𝑅1‘suc 𝐵)) |
16 | suceq 5707 | . . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) | |
17 | 16 | fveq2d 6107 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑅1‘suc 𝑥) = (𝑅1‘suc 𝐵)) |
18 | 17 | eleq2d 2673 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc 𝐵))) |
19 | 18 | rspcev 3282 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) |
20 | 7, 15, 19 | syl2anc 691 | . 2 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) |
21 | rankwflemb 8539 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) | |
22 | 20, 21 | sylibr 223 | 1 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 Tr wtr 4680 dom cdm 5038 “ cima 5041 Ord word 5639 Oncon0 5640 Lim wlim 5641 suc csuc 5642 Fun wfun 5798 ‘cfv 5804 𝑅1cr1 8508 |
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-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-pred 5597 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-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-r1 8510 |
This theorem is referenced by: rankr1ai 8544 pwwf 8553 sswf 8554 unwf 8556 uniwf 8565 rankonidlem 8574 r1pw 8591 r1pwcl 8593 rankr1id 8608 tcrank 8630 dfac12lem2 8849 r1limwun 9437 r1wunlim 9438 inatsk 9479 |
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