Theorem naddwordnexlem1 43359

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 equal to or larger than 𝐴. (Contributed by RP, 14-Feb-2025.)

Hypotheses

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)

Assertion

Ref	Expression
naddwordnexlem1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Proof of Theorem naddwordnexlem1

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 43358	. 2 ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵))
8		omelon 9575	. . . . . . 7 ⊢ ω ∈ On
9		onelon 6345	. . . . . . . . 9 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On)
10	4, 3, 9	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On)
11		onsuc 7767	. . . . . . . 8 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
12	10, 11	syl 17	. . . . . . 7 ⊢ (𝜑 → suc 𝐶 ∈ On)
13		omcl 8477	. . . . . . 7 ⊢ ((ω ∈ On ∧ suc 𝐶 ∈ On) → (ω ·o suc 𝐶) ∈ On)
14	8, 12, 13	sylancr 587	. . . . . 6 ⊢ (𝜑 → (ω ·o suc 𝐶) ∈ On)
15		onelss 6362	. . . . . 6 ⊢ ((ω ·o suc 𝐶) ∈ On → (𝐴 ∈ (ω ·o suc 𝐶) → 𝐴 ⊆ (ω ·o suc 𝐶)))
16	14, 15	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) → 𝐴 ⊆ (ω ·o suc 𝐶)))
17	16	adantrd 491	. . . 4 ⊢ (𝜑 → ((𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵) → 𝐴 ⊆ (ω ·o suc 𝐶)))
18	17	imp 406	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) → 𝐴 ⊆ (ω ·o suc 𝐶))
19		simprr 772	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) → (ω ·o suc 𝐶) ⊆ 𝐵)
20	18, 19	sstrd 3954	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) → 𝐴 ⊆ 𝐵)
21	7, 20	mpdan 687	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3911  Oncon0 6320  suc csuc 6322  (class class class)co 7369  ωcom 7822   +_o coa 8408   ·_o comu 8409

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691  ax-inf2 9570

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-oadd 8415  df-omul 8416

This theorem is referenced by:  oawordex3  43362