Theorem oawordex3 43720

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 8487. (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 43717	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
8		omelon 9560	. . . . . . 7 ⊢ ω ∈ On
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → ω ∈ On)
10		onelon 6343	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On)
11	4, 3, 10	syl2anc 585	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ On)
12		omcl 8466	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On)
13	9, 11, 12	syl2anc 585	. . . . 5 ⊢ (𝜑 → (ω ·o 𝐶) ∈ On)
14		nnon 7817	. . . . . 6 ⊢ (𝑀 ∈ ω → 𝑀 ∈ On)
15	5, 14	syl 17	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ On)
16		oacl 8465	. . . . 5 ⊢ (((ω ·o 𝐶) ∈ On ∧ 𝑀 ∈ On) → ((ω ·o 𝐶) +o 𝑀) ∈ On)
17	13, 15, 16	syl2anc 585	. . . 4 ⊢ (𝜑 → ((ω ·o 𝐶) +o 𝑀) ∈ On)
18	1, 17	eqeltrd 2837	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
19		omcl 8466	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·o 𝐷) ∈ On)
20	9, 4, 19	syl2anc 585	. . . . 5 ⊢ (𝜑 → (ω ·o 𝐷) ∈ On)
21	6, 5	jca 511	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω))
22		ontr1 6365	. . . . . . 7 ⊢ (ω ∈ On → ((𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω) → 𝑁 ∈ ω))
23	9, 21, 22	sylc 65	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
24		nnon 7817	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
25	23, 24	syl 17	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ On)
26		oacl 8465	. . . . 5 ⊢ (((ω ·o 𝐷) ∈ On ∧ 𝑁 ∈ On) → ((ω ·o 𝐷) +o 𝑁) ∈ On)
27	20, 25, 26	syl2anc 585	. . . 4 ⊢ (𝜑 → ((ω ·o 𝐷) +o 𝑁) ∈ On)
28	2, 27	eqeltrd 2837	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
29		oawordex 8487	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
30	18, 28, 29	syl2anc 585	. 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 1542   ∈ wcel 2114  ∃wrex 3061   ⊆ wss 3902  Oncon0 6318  (class class class)co 7361  ωcom 7811   +_o coa 8397   ·_o comu 8398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7683  ax-inf2 9555

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-2nd 7937  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-oadd 8404  df-omul 8405

This theorem is referenced by: (None)