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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 8594. (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 43387 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
8 | omelon 9684 | . . . . . . 7 ⊢ ω ∈ On | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ω ∈ On) |
10 | onelon 6411 | . . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On) | |
11 | 4, 3, 10 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ On) |
12 | omcl 8573 | . . . . . 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 8572 | . . . . 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 2839 | . . 3 ⊢ (𝜑 → 𝐴 ∈ On) |
19 | omcl 8573 | . . . . . 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 6432 | . . . . . . 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 8572 | . . . . 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 2839 | . . 3 ⊢ (𝜑 → 𝐵 ∈ On) |
29 | oawordex 8594 | . . 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 1537 ∈ wcel 2106 ∃wrex 3068 ⊆ wss 3963 Oncon0 6386 (class class class)co 7431 ωcom 7887 +o coa 8502 ·o comu 8503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-oadd 8509 df-omul 8510 |
This theorem is referenced by: (None) |
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