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Mirrors > Home > MPE Home > Th. List > rankr1clem | Structured version Visualization version GIF version |
Description: Lemma for rankr1c 9845. (Contributed by NM, 6-Oct-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
rankr1clem | ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐵 ⊆ (rank‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankr1ag 9826 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐴 ∈ (𝑅1‘𝐵) ↔ (rank‘𝐴) ∈ 𝐵)) | |
2 | 1 | notbid 318 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ ¬ (rank‘𝐴) ∈ 𝐵)) |
3 | r1funlim 9790 | . . . . . . 7 ⊢ (Fun 𝑅1 ∧ Lim dom 𝑅1) | |
4 | 3 | simpri 485 | . . . . . 6 ⊢ Lim dom 𝑅1 |
5 | limord 6429 | . . . . . 6 ⊢ (Lim dom 𝑅1 → Ord dom 𝑅1) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ Ord dom 𝑅1 |
7 | ordelon 6393 | . . . . 5 ⊢ ((Ord dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On) | |
8 | 6, 7 | mpan 689 | . . . 4 ⊢ (𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On) |
9 | 8 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → 𝐵 ∈ On) |
10 | rankon 9819 | . . 3 ⊢ (rank‘𝐴) ∈ On | |
11 | ontri1 6403 | . . 3 ⊢ ((𝐵 ∈ On ∧ (rank‘𝐴) ∈ On) → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ 𝐵)) | |
12 | 9, 10, 11 | sylancl 585 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (𝐵 ⊆ (rank‘𝐴) ↔ ¬ (rank‘𝐴) ∈ 𝐵)) |
13 | 2, 12 | bitr4d 282 | 1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ∈ dom 𝑅1) → (¬ 𝐴 ∈ (𝑅1‘𝐵) ↔ 𝐵 ⊆ (rank‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2099 ⊆ wss 3947 ∪ cuni 4908 dom cdm 5678 “ cima 5681 Ord word 6368 Oncon0 6369 Lim wlim 6370 Fun wfun 6542 ‘cfv 6548 𝑅1cr1 9786 rankcrnk 9787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 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 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-r1 9788 df-rank 9789 |
This theorem is referenced by: rankr1c 9845 ssrankr1 9859 |
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