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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 9804 | . . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
2 | 1 | simpri 485 | . . . . 5 ⊢ Lim dom 𝑅1 |
3 | limord 6446 | . . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
4 | ordsson 7802 | . . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On) | |
5 | 2, 3, 4 | mp2b 10 | . . . 4 ⊢ dom 𝑅1 ⊆ On |
6 | elfvdm 6944 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ dom 𝑅1) | |
7 | 5, 6 | sselid 3993 | . . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐵 ∈ On) |
8 | r1tr 9814 | . . . . . 6 ⊢ Tr (𝑅1‘𝐵) | |
9 | trss 5276 | . . . . . 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 4608 | . . . . 5 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → (𝐴 ∈ 𝒫 (𝑅1‘𝐵) ↔ 𝐴 ⊆ (𝑅1‘𝐵))) | |
12 | 10, 11 | mpbird 257 | . . . 4 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ 𝒫 (𝑅1‘𝐵)) |
13 | r1sucg 9807 | . . . . 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 2842 | . . 3 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ (𝑅1‘suc 𝐵)) |
16 | suceq 6452 | . . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) | |
17 | 16 | fveq2d 6911 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑅1‘suc 𝑥) = (𝑅1‘suc 𝐵)) |
18 | 17 | eleq2d 2825 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 ∈ (𝑅1‘suc 𝑥) ↔ 𝐴 ∈ (𝑅1‘suc 𝐵))) |
19 | 18 | rspcev 3622 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ (𝑅1‘suc 𝐵)) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) |
20 | 7, 15, 19 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) |
21 | rankwflemb 9831 | . 2 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) ↔ ∃𝑥 ∈ On 𝐴 ∈ (𝑅1‘suc 𝑥)) | |
22 | 20, 21 | sylibr 234 | 1 ⊢ (𝐴 ∈ (𝑅1‘𝐵) → 𝐴 ∈ ∪ (𝑅1 “ On)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 𝒫 cpw 4605 ∪ cuni 4912 Tr wtr 5265 dom cdm 5689 “ cima 5692 Ord word 6385 Oncon0 6386 Lim wlim 6387 suc csuc 6388 Fun wfun 6557 ‘cfv 6563 𝑅1cr1 9800 |
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-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-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 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-iun 4998 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-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 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-ov 7434 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-r1 9802 |
This theorem is referenced by: rankr1ai 9836 pwwf 9845 sswf 9846 unwf 9848 uniwf 9857 rankonidlem 9866 r1pw 9883 r1pwcl 9885 rankr1id 9900 tcrank 9922 dfac12lem2 10183 r1limwun 10774 r1wunlim 10775 inatsk 10816 |
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