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Mirrors > Home > MPE Home > Th. List > onwf | Structured version Visualization version GIF version |
Description: The ordinals are all well-founded. (Contributed by Mario Carneiro, 22-Mar-2013.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
onwf | ⊢ On ⊆ ∪ (𝑅1 “ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1fnon 8990 | . . 3 ⊢ 𝑅1 Fn On | |
2 | fndm 6288 | . . 3 ⊢ (𝑅1 Fn On → dom 𝑅1 = On) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ dom 𝑅1 = On |
4 | rankonidlem 9051 | . . . 4 ⊢ (𝑥 ∈ dom 𝑅1 → (𝑥 ∈ ∪ (𝑅1 “ On) ∧ (rank‘𝑥) = 𝑥)) | |
5 | 4 | simpld 487 | . . 3 ⊢ (𝑥 ∈ dom 𝑅1 → 𝑥 ∈ ∪ (𝑅1 “ On)) |
6 | 5 | ssriv 3862 | . 2 ⊢ dom 𝑅1 ⊆ ∪ (𝑅1 “ On) |
7 | 3, 6 | eqsstr3i 3892 | 1 ⊢ On ⊆ ∪ (𝑅1 “ On) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2050 ⊆ wss 3829 ∪ cuni 4712 dom cdm 5407 “ cima 5410 Oncon0 6029 Fn wfn 6183 ‘cfv 6188 𝑅1cr1 8985 rankcrnk 8986 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-om 7397 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-r1 8987 df-rank 8988 |
This theorem is referenced by: dfac12r 9366 r1tskina 10002 |
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