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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 9807 | . . . . . 6 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
| 2 | 1 | simpri 485 | . . . . 5 ⊢ Lim dom 𝑅1 | 
| 3 | limord 6443 | . . . . 5 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
| 4 | ordsson 7804 | . . . . 5 ⊢ (Ord dom 𝑅1 → dom 𝑅1 ⊆ On) | |
| 5 | 2, 3, 4 | mp2b 10 | . . . 4 ⊢ dom 𝑅1 ⊆ On | 
| 6 | 5 | sseli 3978 | . . 3 ⊢ (𝐴 ∈ dom 𝑅1 → 𝐴 ∈ On) | 
| 7 | 5 | sseli 3978 | . . 3 ⊢ (𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On) | 
| 8 | onsseleq 6424 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | |
| 9 | 6, 7, 8 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ 𝐵 ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵))) | 
| 10 | r1tr 9817 | . . . 4 ⊢ Tr (𝑅1‘𝐵) | |
| 11 | r1ordg 9819 | . . . . 5 ⊢ (𝐵 ∈ dom 𝑅1 → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))) | |
| 12 | 11 | adantl 481 | . . . 4 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ 𝐵 → (𝑅1‘𝐴) ∈ (𝑅1‘𝐵))) | 
| 13 | trss 5269 | . . . 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 6905 | . . . . 5 ⊢ (𝐴 = 𝐵 → (𝑅1‘𝐴) = (𝑅1‘𝐵)) | |
| 16 | eqimss 4041 | . . . . 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 859 | . 2 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵) → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) | 
| 20 | 9, 19 | sylbid 240 | 1 ⊢ ((𝐴 ∈ dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ⊆ 𝐵 → (𝑅1‘𝐴) ⊆ (𝑅1‘𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 Tr wtr 5258 dom cdm 5684 Ord word 6382 Oncon0 6383 Lim wlim 6384 Fun wfun 6554 ‘cfv 6560 𝑅1cr1 9803 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-r1 9805 | 
| This theorem is referenced by: r1ord3 9823 r1val1 9827 rankr1ag 9843 unwf 9851 rankelb 9865 rankonidlem 9869 | 
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