Theorem naddwordnexlem1 44400

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 44399	. 2 ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵))
8		omelon 9648	. . . . . . 7 ⊢ ω ∈ On
9		onelon 6388	. . . . . . . . 9 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On)
10	4, 3, 9	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On)
11		onsuc 7825	. . . . . . . 8 ⊢ (𝐶 ∈ On → suc 𝐶 ∈ On)
12	10, 11	syl 18	. . . . . . 7 ⊢ (𝜑 → suc 𝐶 ∈ On)
13		omcl 8545	. . . . . . 7 ⊢ ((ω ∈ On ∧ suc 𝐶 ∈ On) → (ω ·o suc 𝐶) ∈ On)
14	8, 12, 13	sylancr 599	. . . . . 6 ⊢ (𝜑 → (ω ·o suc 𝐶) ∈ On)
15		onelss 6406	. . . . . 6 ⊢ ((ω ·o suc 𝐶) ∈ On → (𝐴 ∈ (ω ·o suc 𝐶) → 𝐴 ⊆ (ω ·o suc 𝐶)))
16	14, 15	syl 18	. . . . 5 ⊢ (𝜑 → (𝐴 ∈ (ω ·o suc 𝐶) → 𝐴 ⊆ (ω ·o suc 𝐶)))
17	16	adantrd 497	. . . 4 ⊢ (𝜑 → ((𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵) → 𝐴 ⊆ (ω ·o suc 𝐶)))
18	17	imp 412	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) → 𝐴 ⊆ (ω ·o suc 𝐶))
19		simprr 785	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) → (ω ·o suc 𝐶) ⊆ 𝐵)
20	18, 19	sstrd 3939	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ (ω ·o suc 𝐶) ∧ (ω ·o suc 𝐶) ⊆ 𝐵)) → 𝐴 ⊆ 𝐵)
21	7, 20	mpdan 700	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3914  Oncon0 6363  suc csuc 6365  (class class class)co 7421  ωcom 7878   +_o coa 8474   ·_o comu 8475

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7752  ax-inf2 9643

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  df-pss 4035  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-2nd 8003  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-oadd 8481  df-omul 8482

This theorem is used by:  oawordex3  44403