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Mirrors > Home > MPE Home > Th. List > Mathboxes > oawordex3 | Structured version Visualization version GIF version |
Description: When 𝐴 is the sum of a limit ordinal (or zero) and a natural number and 𝐵 is the sum of a larger limit ordinal and a smaller natural number, some ordinal sum of 𝐴 is equal to 𝐵. This is a specialization of oawordex 8552. (Contributed by RP, 14-Feb-2025.) |
Ref | Expression |
---|---|
naddwordnex.a | ⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀)) |
naddwordnex.b | ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁)) |
naddwordnex.c | ⊢ (𝜑 → 𝐶 ∈ 𝐷) |
naddwordnex.d | ⊢ (𝜑 → 𝐷 ∈ On) |
naddwordnex.m | ⊢ (𝜑 → 𝑀 ∈ ω) |
naddwordnex.n | ⊢ (𝜑 → 𝑁 ∈ 𝑀) |
Ref | Expression |
---|---|
oawordex3 | ⊢ (𝜑 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | naddwordnex.a | . . 3 ⊢ (𝜑 → 𝐴 = ((ω ·o 𝐶) +o 𝑀)) | |
2 | naddwordnex.b | . . 3 ⊢ (𝜑 → 𝐵 = ((ω ·o 𝐷) +o 𝑁)) | |
3 | naddwordnex.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐷) | |
4 | naddwordnex.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ On) | |
5 | naddwordnex.m | . . 3 ⊢ (𝜑 → 𝑀 ∈ ω) | |
6 | naddwordnex.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ 𝑀) | |
7 | 1, 2, 3, 4, 5, 6 | naddwordnexlem1 42637 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
8 | omelon 9637 | . . . . . . 7 ⊢ ω ∈ On | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ω ∈ On) |
10 | onelon 6379 | . . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On) | |
11 | 4, 3, 10 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ On) |
12 | omcl 8531 | . . . . . 6 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On) | |
13 | 9, 11, 12 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (ω ·o 𝐶) ∈ On) |
14 | nnon 7854 | . . . . . 6 ⊢ (𝑀 ∈ ω → 𝑀 ∈ On) | |
15 | 5, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ On) |
16 | oacl 8530 | . . . . 5 ⊢ (((ω ·o 𝐶) ∈ On ∧ 𝑀 ∈ On) → ((ω ·o 𝐶) +o 𝑀) ∈ On) | |
17 | 13, 15, 16 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ((ω ·o 𝐶) +o 𝑀) ∈ On) |
18 | 1, 17 | eqeltrd 2825 | . . 3 ⊢ (𝜑 → 𝐴 ∈ On) |
19 | omcl 8531 | . . . . . 6 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·o 𝐷) ∈ On) | |
20 | 9, 4, 19 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (ω ·o 𝐷) ∈ On) |
21 | 6, 5 | jca 511 | . . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω)) |
22 | ontr1 6400 | . . . . . . 7 ⊢ (ω ∈ On → ((𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω) → 𝑁 ∈ ω)) | |
23 | 9, 21, 22 | sylc 65 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω) |
24 | nnon 7854 | . . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On) | |
25 | 23, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ On) |
26 | oacl 8530 | . . . . 5 ⊢ (((ω ·o 𝐷) ∈ On ∧ 𝑁 ∈ On) → ((ω ·o 𝐷) +o 𝑁) ∈ On) | |
27 | 20, 25, 26 | syl2anc 583 | . . . 4 ⊢ (𝜑 → ((ω ·o 𝐷) +o 𝑁) ∈ On) |
28 | 2, 27 | eqeltrd 2825 | . . 3 ⊢ (𝜑 → 𝐵 ∈ On) |
29 | oawordex 8552 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)) | |
30 | 18, 28, 29 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)) |
31 | 7, 30 | mpbid 231 | 1 ⊢ (𝜑 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3062 ⊆ wss 3940 Oncon0 6354 (class class class)co 7401 ωcom 7848 +o coa 8458 ·o comu 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 5275 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 ax-inf2 9632 |
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-rmo 3368 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-int 4941 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-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-oadd 8465 df-omul 8466 |
This theorem is referenced by: (None) |
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