Theorem omnord1ex 44567

Description: When omega is multiplied on the right to ordinals one and two, ordering of the products is not equivalent to the ordering of the ordinals on the left. Remark 3.18 of [Schloeder] p. 10. (Contributed by RP, 29-Jan-2025.)

Assertion

Ref	Expression
omnord1ex	⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))

Proof of Theorem omnord1ex

Step	Hyp	Ref	Expression
1		1oelpr 8525	. . . 4 ⊢ 1o ∈ {∅, 1o}
2		df2o3 8522	. . . 4 ⊢ 2o = {∅, 1o}
3	1, 2	eleqtrri 2863	. . 3 ⊢ 1o ∈ 2o
4		ordom 7921	. . . 4 ⊢ Ord ω
5		ordirr 6406	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
6		omelon 9684	. . . . . . 7 ⊢ ω ∈ On
7		1onn 8687	. . . . . . 7 ⊢ 1o ∈ ω
8		0lt1o 8550	. . . . . . 7 ⊢ ∅ ∈ 1o
9		omabslem 8697	. . . . . . 7 ⊢ ((ω ∈ On ∧ 1o ∈ ω ∧ ∅ ∈ 1o) → (1o ·o ω) = ω)
10	6, 7, 8, 9	mp3an 1491	. . . . . 6 ⊢ (1o ·o ω) = ω
11		2omomeqom 44566	. . . . . 6 ⊢ (2o ·o ω) = ω
12	10, 11	eleq12i 2857	. . . . 5 ⊢ ((1o ·o ω) ∈ (2o ·o ω) ↔ ω ∈ ω)
13	5, 12	sylnibr 332	. . . 4 ⊢ (Ord ω → ¬ (1o ·o ω) ∈ (2o ·o ω))
14	4, 13	ax-mp 5	. . 3 ⊢ ¬ (1o ·o ω) ∈ (2o ·o ω)
15	3, 14	2th 267	. 2 ⊢ (1o ∈ 2o ↔ ¬ (1o ·o ω) ∈ (2o ·o ω))
16		xor3 385	. 2 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) ↔ (1o ∈ 2o ↔ ¬ (1o ·o ω) ∈ (2o ·o ω)))
17	15, 16	mpbir 234	1 ⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2147  ∅c0 4281  {cpr 4590  Ord word 6387  Oncon0 6388  (class class class)co 7451  ωcom 7911  1_oc1o 8507  2_oc2o 8508   ·_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-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-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-1o 8514  df-2o 8515  df-oadd 8518  df-omul 8519

This theorem is used by:  omnord1  44568