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Theorem oawordex3 43390
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 8594. (Contributed by RP, 14-Feb-2025.)
Hypotheses
Ref Expression
naddwordnex.a (𝜑𝐴 = ((ω ·o 𝐶) +o 𝑀))
naddwordnex.b (𝜑𝐵 = ((ω ·o 𝐷) +o 𝑁))
naddwordnex.c (𝜑𝐶𝐷)
naddwordnex.d (𝜑𝐷 ∈ On)
naddwordnex.m (𝜑𝑀 ∈ ω)
naddwordnex.n (𝜑𝑁𝑀)
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
StepHypRef Expression
1 naddwordnex.a . . 3 (𝜑𝐴 = ((ω ·o 𝐶) +o 𝑀))
2 naddwordnex.b . . 3 (𝜑𝐵 = ((ω ·o 𝐷) +o 𝑁))
3 naddwordnex.c . . 3 (𝜑𝐶𝐷)
4 naddwordnex.d . . 3 (𝜑𝐷 ∈ On)
5 naddwordnex.m . . 3 (𝜑𝑀 ∈ ω)
6 naddwordnex.n . . 3 (𝜑𝑁𝑀)
71, 2, 3, 4, 5, 6naddwordnexlem1 43387 . 2 (𝜑𝐴𝐵)
8 omelon 9684 . . . . . . 7 ω ∈ On
98a1i 11 . . . . . 6 (𝜑 → ω ∈ On)
10 onelon 6411 . . . . . . 7 ((𝐷 ∈ On ∧ 𝐶𝐷) → 𝐶 ∈ On)
114, 3, 10syl2anc 584 . . . . . 6 (𝜑𝐶 ∈ On)
12 omcl 8573 . . . . . 6 ((ω ∈ On ∧ 𝐶 ∈ On) → (ω ·o 𝐶) ∈ On)
139, 11, 12syl2anc 584 . . . . 5 (𝜑 → (ω ·o 𝐶) ∈ On)
14 nnon 7893 . . . . . 6 (𝑀 ∈ ω → 𝑀 ∈ On)
155, 14syl 17 . . . . 5 (𝜑𝑀 ∈ On)
16 oacl 8572 . . . . 5 (((ω ·o 𝐶) ∈ On ∧ 𝑀 ∈ On) → ((ω ·o 𝐶) +o 𝑀) ∈ On)
1713, 15, 16syl2anc 584 . . . 4 (𝜑 → ((ω ·o 𝐶) +o 𝑀) ∈ On)
181, 17eqeltrd 2839 . . 3 (𝜑𝐴 ∈ On)
19 omcl 8573 . . . . . 6 ((ω ∈ On ∧ 𝐷 ∈ On) → (ω ·o 𝐷) ∈ On)
209, 4, 19syl2anc 584 . . . . 5 (𝜑 → (ω ·o 𝐷) ∈ On)
216, 5jca 511 . . . . . . 7 (𝜑 → (𝑁𝑀𝑀 ∈ ω))
22 ontr1 6432 . . . . . . 7 (ω ∈ On → ((𝑁𝑀𝑀 ∈ ω) → 𝑁 ∈ ω))
239, 21, 22sylc 65 . . . . . 6 (𝜑𝑁 ∈ ω)
24 nnon 7893 . . . . . 6 (𝑁 ∈ ω → 𝑁 ∈ On)
2523, 24syl 17 . . . . 5 (𝜑𝑁 ∈ On)
26 oacl 8572 . . . . 5 (((ω ·o 𝐷) ∈ On ∧ 𝑁 ∈ On) → ((ω ·o 𝐷) +o 𝑁) ∈ On)
2720, 25, 26syl2anc 584 . . . 4 (𝜑 → ((ω ·o 𝐷) +o 𝑁) ∈ On)
282, 27eqeltrd 2839 . . 3 (𝜑𝐵 ∈ On)
29 oawordex 8594 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
3018, 28, 29syl2anc 584 . 2 (𝜑 → (𝐴𝐵 ↔ ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵))
317, 30mpbid 232 1 (𝜑 → ∃𝑥 ∈ On (𝐴 +o 𝑥) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  wss 3963  Oncon0 6386  (class class class)co 7431  ωcom 7887   +o coa 8502   ·o comu 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509  df-omul 8510
This theorem is referenced by: (None)
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