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Mirrors > Home > MPE Home > Th. List > Mathboxes > oaltublim | Structured version Visualization version GIF version |
Description: Given 𝐶 is a limit ordinal, the sum of any ordinal with an ordinal less than 𝐶 is less than the sum of the first ordinal with 𝐶. Lemma 3.5 of [Schloeder] p. 7. (Contributed by RP, 29-Jan-2025.) |
Ref | Expression |
---|---|
oaltublim | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐶 ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limord 6414 | . . . . . 6 ⊢ (Lim 𝐶 → Ord 𝐶) | |
2 | elex 3485 | . . . . . 6 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
3 | 1, 2 | anim12i 612 | . . . . 5 ⊢ ((Lim 𝐶 ∧ 𝐶 ∈ 𝑉) → (Ord 𝐶 ∧ 𝐶 ∈ V)) |
4 | elon2 6365 | . . . . 5 ⊢ (𝐶 ∈ On ↔ (Ord 𝐶 ∧ 𝐶 ∈ V)) | |
5 | 3, 4 | sylibr 233 | . . . 4 ⊢ ((Lim 𝐶 ∧ 𝐶 ∈ 𝑉) → 𝐶 ∈ On) |
6 | 5 | 3ad2ant3 1132 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐶 ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → 𝐶 ∈ On) |
7 | simp1 1133 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐶 ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → 𝐴 ∈ On) | |
8 | 6, 7 | jca 511 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐶 ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐶 ∈ On ∧ 𝐴 ∈ On)) |
9 | simp2 1134 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐶 ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → 𝐵 ∈ 𝐶) | |
10 | oaordi 8541 | . 2 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ∈ 𝐶 → (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶))) | |
11 | 8, 9, 10 | sylc 65 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ 𝐶 ∧ (Lim 𝐶 ∧ 𝐶 ∈ 𝑉)) → (𝐴 +o 𝐵) ∈ (𝐴 +o 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 ∈ wcel 2098 Vcvv 3466 Ord word 6353 Oncon0 6354 Lim wlim 6355 (class class class)co 7401 +o coa 8458 |
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 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-oadd 8465 |
This theorem is referenced by: (None) |
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