Theorem oawordex3 43396

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 8524. (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
oawordex3	⊢ (𝜑 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝑀(𝑥)   𝑁(𝑥)

Proof of Theorem oawordex3

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 43393	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
8		omelon 9606	. . . . . . 7 ⊢ ω ∈ On
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → ω ∈ On)
10		onelon 6360	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On)
11	4, 3, 10	syl2anc 584	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ On)
12		omcl 8503	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On)
13	9, 11, 12	syl2anc 584	. . . . 5 ⊢ (𝜑 → (ω ·o 𝐶) ∈ On)
14		nnon 7851	. . . . . 6 ⊢ (𝑀 ∈ ω → 𝑀 ∈ On)
15	5, 14	syl 17	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ On)
16		oacl 8502	. . . . 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 2829	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
19		omcl 8503	. . . . . 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 6382	. . . . . . 7 ⊢ (ω ∈ On → ((𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω) → 𝑁 ∈ ω))
23	9, 21, 22	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
24		nnon 7851	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
25	23, 24	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ On)
26		oacl 8502	. . . . 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 2829	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
29		oawordex 8524	. . 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)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3054   ⊆ wss 3917  Oncon0 6335  (class class class)co 7390  ωcom 7845   +_o coa 8434   ·_o comu 8435

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714  ax-inf2 9601

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-oadd 8441  df-omul 8442

This theorem is referenced by: (None)