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Mirrors > Home > MPE Home > Th. List > r1ord3g | Structured version Visualization version GIF version |
Description: Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.) |
Ref | Expression |
---|---|
r1ord3g | ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1funlim 9183 | . . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
2 | 1 | simpri 486 | . . . . 5 ⊢ Lim dom 𝑅1 |
3 | limord 6243 | . . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
4 | ordsson 7493 | . . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On) | |
5 | 2, 3, 4 | mp2b 10 | . . . 4 ⊢ dom 𝑅1 ⊆ On |
6 | 5 | sseli 3960 | . . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On) |
7 | 5 | sseli 3960 | . . 3 ⊢ (𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On) |
8 | onsseleq 6225 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
9 | 6, 7, 8 | syl2an 595 | . 2 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
10 | r1tr 9193 | . . . 4 ⊢ Tr (𝑅1‘𝐵) | |
11 | r1ordg 9195 | . . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))) | |
12 | 11 | adantl 482 | . . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))) |
13 | trss 5172 | . . . 4 ⊢ (Tr (𝑅1‘𝐵) → ((𝑅1‘𝐴) ∈ (𝑅1‘𝐵) → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) | |
14 | 10, 12, 13 | mpsylsyld 69 | . . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) |
15 | fveq2 6663 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑅1‘𝐴) = (𝑅1‘𝐵)) | |
16 | eqimss 4020 | . . . . 5 ⊢ ((𝑅1‘𝐴) = (𝑅1‘𝐵) → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵)) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵)) |
18 | 17 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 = 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) |
19 | 14, 18 | jaod 853 | . 2 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) |
20 | 9, 19 | sylbid 241 | 1 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 Tr wtr 5163 dom cdm 5548 Ord word 6183 Oncon0 6184 Lim wlim 6185 Fun wfun 6342 ‘cfv 6348 𝑅1cr1 9179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-r1 9181 |
This theorem is referenced by: r1ord3 9199 r1val1 9203 rankr1ag 9219 unwf 9227 rankelb 9241 rankonidlem 9245 |
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