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Mirrors > Home > MPE Home > Th. List > Mathboxes > nadd2rabord | Structured version Visualization version GIF version |
Description: The set of ordinals which have a natural sum less than some ordinal is an ordinal. (Contributed by RP, 20-Dec-2024.) |
Ref | Expression |
---|---|
nadd2rabord | ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Ord {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4072 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ⊆ 𝐴 | |
2 | ordsson 7767 | . . . 4 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
3 | 2 | 3ad2ant1 1130 | . . 3 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → 𝐴 ⊆ On) |
4 | 1, 3 | sstrid 3988 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ⊆ On) |
5 | nadd2rabtr 42715 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Tr {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}) | |
6 | dford5 7768 | . 2 ⊢ (Ord {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ↔ ({𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶} ⊆ On ∧ Tr {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶})) | |
7 | 4, 5, 6 | sylanbrc 582 | 1 ⊢ ((Ord 𝐴 ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → Ord {𝑥 ∈ 𝐴 ∣ (𝐵 +no 𝑥) ∈ 𝐶}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 ∈ wcel 2098 {crab 3426 ⊆ wss 3943 Tr wtr 5258 Ord word 6357 Oncon0 6358 (class class class)co 7405 +no cnadd 8666 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-frecs 8267 df-nadd 8667 |
This theorem is referenced by: nadd2rabon 42718 |
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