![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
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 43344 | . 2 ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) |
8 | ssel 3989 | . . 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 1535 ∈ wcel 2104 ⊆ wss 3963 Oncon0 6380 suc csuc 6382 (class class class)co 7425 ωcom 7880 +o coa 8496 ·o comu 8497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5430 ax-un 7747 ax-inf2 9672 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3377 df-rab 3433 df-v 3479 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 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 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 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-ov 7428 df-oprab 7429 df-mpo 7430 df-om 7881 df-2nd 8008 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-oadd 8503 df-omul 8504 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |