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Mirrors > Home > MPE Home > Th. List > Mathboxes > oasubex | Structured version Visualization version GIF version |
Description: While subtraction can't be a binary operation on ordinals, for any pair of ordinals there exists an ordinal that can be added to the lessor (or equal) one which will sum to the greater. Theorem 2.19 of [Schloeder] p. 6. (Contributed by RP, 29-Jan-2025.) |
Ref | Expression |
---|---|
oasubex | ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → ∃𝑐 ∈ On (𝑐 ⊆ 𝐴 ∧ (𝐵 +o 𝑐) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1137 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ On) | |
2 | simp1 1136 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → 𝐴 ∈ On) | |
3 | simp3 1138 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐴) | |
4 | oawordex 8542 | . . . 4 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 ⊆ 𝐴 ↔ ∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴)) | |
5 | 4 | biimpa 477 | . . 3 ⊢ (((𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ 𝐵 ⊆ 𝐴) → ∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴) |
6 | 1, 2, 3, 5 | syl21anc 836 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → ∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴) |
7 | simpr 485 | . . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → (𝐵 +o 𝑐) = 𝐴) | |
8 | simpl1 1191 | . . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → 𝐴 ∈ On) | |
9 | simpl2 1192 | . . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → 𝐵 ∈ On) | |
10 | oaword2 8538 | . . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴)) | |
11 | 8, 9, 10 | syl2anc 584 | . . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → 𝐴 ⊆ (𝐵 +o 𝐴)) |
12 | 11 | adantr 481 | . . . . . . 7 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → 𝐴 ⊆ (𝐵 +o 𝐴)) |
13 | 7, 12 | eqsstrd 4017 | . . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴)) |
14 | simpr 485 | . . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → 𝑐 ∈ On) | |
15 | oaword 8534 | . . . . . . . 8 ⊢ ((𝑐 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑐 ⊆ 𝐴 ↔ (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴))) | |
16 | 14, 8, 9, 15 | syl3anc 1371 | . . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → (𝑐 ⊆ 𝐴 ↔ (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴))) |
17 | 16 | adantr 481 | . . . . . 6 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → (𝑐 ⊆ 𝐴 ↔ (𝐵 +o 𝑐) ⊆ (𝐵 +o 𝐴))) |
18 | 13, 17 | mpbird 256 | . . . . 5 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) ∧ (𝐵 +o 𝑐) = 𝐴) → 𝑐 ⊆ 𝐴) |
19 | 18 | ex 413 | . . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → ((𝐵 +o 𝑐) = 𝐴 → 𝑐 ⊆ 𝐴)) |
20 | 19 | ancrd 552 | . . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) ∧ 𝑐 ∈ On) → ((𝐵 +o 𝑐) = 𝐴 → (𝑐 ⊆ 𝐴 ∧ (𝐵 +o 𝑐) = 𝐴))) |
21 | 20 | reximdva 3168 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → (∃𝑐 ∈ On (𝐵 +o 𝑐) = 𝐴 → ∃𝑐 ∈ On (𝑐 ⊆ 𝐴 ∧ (𝐵 +o 𝑐) = 𝐴))) |
22 | 6, 21 | mpd 15 | 1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐵 ⊆ 𝐴) → ∃𝑐 ∈ On (𝑐 ⊆ 𝐴 ∧ (𝐵 +o 𝑐) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ⊆ wss 3945 Oncon0 6354 (class class class)co 7394 +o coa 8447 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 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 7397 df-oprab 7398 df-mpo 7399 df-om 7840 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-oadd 8454 |
This theorem is referenced by: (None) |
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