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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 9795 | . . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
2 | 1 | simpri 484 | . . . . 5 ⊢ Lim dom 𝑅1 |
3 | limord 6432 | . . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
4 | ordsson 7789 | . . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On) | |
5 | 2, 3, 4 | mp2b 10 | . . . 4 ⊢ dom 𝑅1 ⊆ On |
6 | 5 | sseli 3976 | . . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On) |
7 | 5 | sseli 3976 | . . 3 ⊢ (𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On) |
8 | onsseleq 6413 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
9 | 6, 7, 8 | syl2an 594 | . 2 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) |
10 | r1tr 9805 | . . . 4 ⊢ Tr (𝑅1‘𝐵) | |
11 | r1ordg 9807 | . . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))) | |
12 | 11 | adantl 480 | . . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))) |
13 | trss 5278 | . . . 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 6900 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑅1‘𝐴) = (𝑅1‘𝐵)) | |
16 | eqimss 4038 | . . . . 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 857 | . 2 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) |
20 | 9, 19 | sylbid 239 | 1 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ⊆ wss 3947 Tr wtr 5267 dom cdm 5680 Ord word 6371 Oncon0 6372 Lim wlim 6373 Fun wfun 6545 ‘cfv 6551 𝑅1cr1 9791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-om 7875 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-r1 9793 |
This theorem is referenced by: r1ord3 9811 r1val1 9815 rankr1ag 9831 unwf 9839 rankelb 9853 rankonidlem 9857 |
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