Theorem oawordex3 44663

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 8603. (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 44660	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
8		omelon 9684	. . . . . . 7 ⊢ ω ∈ On
9	8	a1i 11	. . . . . 6 ⊢ (𝜑 → ω ∈ On)
10		onelon 6413	. . . . . . 7 ⊢ ((𝐷 ∈ On ∧ 𝐶 ∈ 𝐷) → 𝐶 ∈ On)
11	4, 3, 10	syl2anc 596	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ On)
12		omcl 8582	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On)
13	9, 11, 12	syl2anc 596	. . . . 5 ⊢ (𝜑 → (ω ·o 𝐶) ∈ On)
14		nnon 7917	. . . . . 6 ⊢ (𝑀 ∈ ω → 𝑀 ∈ On)
15	5, 14	syl 18	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ On)
16		oacl 8581	. . . . 5 ⊢ (((ω ·o 𝐶) ∈ On ∧ 𝑀 ∈ On) → ((ω ·o 𝐶) +o 𝑀) ∈ On)
17	13, 15, 16	syl2anc 596	. . . 4 ⊢ (𝜑 → ((ω ·o 𝐶) +o 𝑀) ∈ On)
18	1, 17	eqeltrd 2864	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
19		omcl 8582	. . . . . 6 ⊢ ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·o 𝐷) ∈ On)
20	9, 4, 19	syl2anc 596	. . . . 5 ⊢ (𝜑 → (ω ·o 𝐷) ∈ On)
21	6, 5	jca 521	. . . . . . 7 ⊢ (𝜑 → (𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω))
22		ontr1 6436	. . . . . . 7 ⊢ (ω ∈ On → ((𝑁 ∈ 𝑀 ∧ 𝑀 ∈ ω) → 𝑁 ∈ ω))
23	9, 21, 22	sylc 66	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ω)
24		nnon 7917	. . . . . 6 ⊢ (𝑁 ∈ ω → 𝑁 ∈ On)
25	23, 24	syl 18	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ On)
26		oacl 8581	. . . . 5 ⊢ (((ω ·o 𝐷) ∈ On ∧ 𝑁 ∈ On) → ((ω ·o 𝐷) +o 𝑁) ∈ On)
27	20, 25, 26	syl2anc 596	. . . 4 ⊢ (𝜑 → ((ω ·o 𝐷) +o 𝑁) ∈ On)
28	2, 27	eqeltrd 2864	. . 3 ⊢ (𝜑 → 𝐵 ∈ On)
29		oawordex 8603	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
30	18, 28, 29	syl2anc 596	. 2 ⊢ (𝜑 → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
31	7, 30	mpbid 235	1 ⊢ (𝜑 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∃wrex 3090   ⊆ wss 3900  Oncon0 6388  (class class class)co 7451  ωcom 7911   +_o coa 8511   ·_o comu 8512

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783  ax-inf2 9679

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-oadd 8518  df-omul 8519

This theorem is used by: (None)