Theorem 1oaomeqom 43875

Description: Ordinal one plus omega is equal to omega. See oaabs 8618 for the sum of any natural number on the left and ordinal at least as large as omega on the right. Lemma 3.8 of [Schloeder] p. 8. See oaabs2 8619 where a power of omega is the upper bound of the left and a lower bound on the right. (Contributed by RP, 29-Jan-2025.)

Assertion

Ref	Expression
1oaomeqom	⊢ (1o +o ω) = ω

Proof of Theorem 1oaomeqom

Step	Hyp	Ref	Expression
1		omelon 9599	. 2 ⊢ ω ∈ On
2		1onn 8610	. 2 ⊢ 1o ∈ ω
3		oaabslem 8617	. 2 ⊢ ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω)
4	1, 2, 3	mp2an 702	1 ⊢ (1o +o ω) = ω

Syntax hints:   = wceq 1561   ∈ wcel 2143  Oncon0 6346  (class class class)co 7396  ωcom 7846  1_oc1o 8430   +_o coa 8434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7718  ax-inf2 9594

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-oadd 8441

This theorem is referenced by:  oaabsb  43876  oaordnrex  43877