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Mirrors > Home > MPE Home > Th. List > Mathboxes > naddwordnexlem2 | 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, 𝐵 is larger than 𝐴. (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 |
---|---|
naddwordnexlem2 | ⊢ (𝜑 → 𝐴 ∈ 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 | naddwordnexlem0 43387 | . 2 ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) |
8 | ssel 3976 | . . 3 ⊢ ((ω ·o suc 𝐶) ⊆ 𝐵 → (𝐴 ∈ (ω ·o suc 𝐶) → 𝐴 ∈ 𝐵)) | |
9 | 8 | impcom 407 | . 2 ⊢ ((𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵) → 𝐴 ∈ 𝐵) |
10 | 7, 9 | syl 17 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3950 Oncon0 6382 suc csuc 6384 (class class class)co 7429 ωcom 7883 +o coa 8499 ·o comu 8500 |
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 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pr 5430 ax-un 7751 ax-inf2 9677 |
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 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-ov 7432 df-oprab 7433 df-mpo 7434 df-om 7884 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-oadd 8506 df-omul 8507 |
This theorem is referenced by: (None) |
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