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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 8595. (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 43410 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| 8 | omelon 9686 | . . . . . . 7 ⊢ ω ∈ On | |
| 9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ω ∈ On) |
| 10 | onelon 6409 | . . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On) | |
| 11 | 4, 3, 10 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ On) |
| 12 | omcl 8574 | . . . . . 6 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On) | |
| 13 | 9, 11, 12 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (ω ·o 𝐶) ∈ On) |
| 14 | nnon 7893 | . . . . . 6 ⊢ (𝑀 ∈ ω → 𝑀 ∈ On) | |
| 15 | 5, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ On) |
| 16 | oacl 8573 | . . . . 5 ⊢ (((ω ·o 𝐶) ∈ On ∧ 𝑀 ∈ On) → ((ω ·o 𝐶) +o 𝑀) ∈ On) | |
| 17 | 13, 15, 16 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((ω ·o 𝐶) +o 𝑀) ∈ On) |
| 18 | 1, 17 | eqeltrd 2841 | . . 3 ⊢ (𝜑 → 𝐴 ∈ On) |
| 19 | omcl 8574 | . . . . . 6 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·o 𝐷) ∈ On) | |
| 20 | 9, 4, 19 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (ω ·o 𝐷) ∈ On) |
| 21 | 6, 5 | jca 511 | . . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω)) |
| 22 | ontr1 6430 | . . . . . . 7 ⊢ (ω ∈ On → ((𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω) → 𝑁 ∈ ω)) | |
| 23 | 9, 21, 22 | sylc 65 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω) |
| 24 | nnon 7893 | . . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On) | |
| 25 | 23, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ On) |
| 26 | oacl 8573 | . . . . 5 ⊢ (((ω ·o 𝐷) ∈ On ∧ 𝑁 ∈ On) → ((ω ·o 𝐷) +o 𝑁) ∈ On) | |
| 27 | 20, 25, 26 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ((ω ·o 𝐷) +o 𝑁) ∈ On) |
| 28 | 2, 27 | eqeltrd 2841 | . . 3 ⊢ (𝜑 → 𝐵 ∈ On) |
| 29 | oawordex 8595 | . . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)) | |
| 30 | 18, 28, 29 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)) |
| 31 | 7, 30 | mpbid 232 | 1 ⊢ (𝜑 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ⊆ wss 3951 Oncon0 6384 (class class class)co 7431 ωcom 7887 +o coa 8503 ·o comu 8504 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-oadd 8510 df-omul 8511 |
| This theorem is referenced by: (None) |
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