Theorem 1oaomeqom 42993

Description: Ordinal one plus omega is equal to omega. See oaabs 8667 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 8668 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 9679	. 2 ⊢ ω ∈ On
2		1onn 8659	. 2 ⊢ 1o ∈ ω
3		oaabslem 8666	. 2 ⊢ ((ω ∈ On ∧ 1o ∈ ω) → (1o +o ω) = ω)
4	1, 2, 3	mp2an 690	1 ⊢ (1o +o ω) = ω

Syntax hints:   = wceq 1534   ∈ wcel 2099  Oncon0 6365  (class class class)co 7413  ωcom 7865  1_oc1o 8478   +_o coa 8482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7735  ax-inf2 9674

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6302  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-oadd 8489

This theorem is referenced by:  oaabsb  42994  oaordnrex  42995