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Mirrors > Home > MPE Home > Th. List > sswf | Structured version Visualization version GIF version |
Description: A subset of a well-founded set is well-founded. (Contributed by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
sswf | ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ ∪ (𝑅1 “ On)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rankidb 9542 | . . 3 ⊢ (𝐴 ∈ ∪ (𝑅1 “ On) → 𝐴 ∈ (𝑅1‘suc (rank‘𝐴))) | |
2 | r1sscl 9527 | . . 3 ⊢ ((𝐴 ∈ (𝑅1‘suc (rank‘𝐴)) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (𝑅1‘suc (rank‘𝐴))) | |
3 | 1, 2 | sylan 579 | . 2 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ (𝑅1‘suc (rank‘𝐴))) |
4 | r1elwf 9538 | . 2 ⊢ (𝐵 ∈ (𝑅1‘suc (rank‘𝐴)) → 𝐵 ∈ ∪ (𝑅1 “ On)) | |
5 | 3, 4 | syl 17 | 1 ⊢ ((𝐴 ∈ ∪ (𝑅1 “ On) ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ ∪ (𝑅1 “ On)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2109 ⊆ wss 3891 ∪ cuni 4844 “ cima 5591 Oncon0 6263 suc csuc 6265 ‘cfv 6430 𝑅1cr1 9504 rankcrnk 9505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-reu 3072 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-ov 7271 df-om 7701 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-r1 9506 df-rank 9507 |
This theorem is referenced by: snwf 9551 unwf 9552 uniwf 9561 rankssb 9590 |
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