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Mirrors > Home > MPE Home > Th. List > oaword2 | Structured version Visualization version GIF version |
Description: An ordinal is less than or equal to its sum with another. Theorem 21 of [Suppes] p. 209. Lemma 3.3 of [Schloeder] p. 7. (Contributed by NM, 7-Dec-2004.) |
Ref | Expression |
---|---|
oaword2 | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ss 4389 | . . 3 ⊢ ∅ ⊆ 𝐵 | |
2 | 0elon 6409 | . . . . 5 ⊢ ∅ ∈ On | |
3 | oawordri 8546 | . . . . 5 ⊢ ((∅ ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐵 → (∅ +o 𝐴) ⊆ (𝐵 +o 𝐴))) | |
4 | 2, 3 | mp3an1 1444 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐵 → (∅ +o 𝐴) ⊆ (𝐵 +o 𝐴))) |
5 | oa0r 8534 | . . . . . 6 ⊢ (𝐴 ∈ On → (∅ +o 𝐴) = 𝐴) | |
6 | 5 | adantl 481 | . . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ +o 𝐴) = 𝐴) |
7 | 6 | sseq1d 4006 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((∅ +o 𝐴) ⊆ (𝐵 +o 𝐴) ↔ 𝐴 ⊆ (𝐵 +o 𝐴))) |
8 | 4, 7 | sylibd 238 | . . 3 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (∅ ⊆ 𝐵 → 𝐴 ⊆ (𝐵 +o 𝐴))) |
9 | 1, 8 | mpi 20 | . 2 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴)) |
10 | 9 | ancoms 458 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ⊆ wss 3941 ∅c0 4315 Oncon0 6355 (class class class)co 7402 +o coa 8459 |
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 5276 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-oadd 8466 |
This theorem is referenced by: oawordeulem 8550 nnarcl 8612 oaabslem 8643 oaabs2 8645 cantnfle 9663 oasubex 42550 naddwordnexlem4 42666 |
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