Theorem omnord1ex 44014

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 8465	. . . 4 ⊢ 1o ∈ {∅, 1o}
2		df2o3 8462	. . . 4 ⊢ 2o = {∅, 1o}
3	1, 2	eleqtrri 2862	. . 3 ⊢ 1o ∈ 2o
4		ordom 7873	. . . 4 ⊢ Ord ω
5		ordirr 6380	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
6		omelon 9616	. . . . . . 7 ⊢ ω ∈ On
7		1onn 8627	. . . . . . 7 ⊢ 1o ∈ ω
8		0lt1o 8490	. . . . . . 7 ⊢ ∅ ∈ 1o
9		omabslem 8637	. . . . . . 7 ⊢ ((ω ∈ On ∧ 1o ∈ ω ∧ ∅ ∈ 1o) → (1o ·o ω) = ω)
10	6, 7, 8, 9	mp3an 1490	. . . . . 6 ⊢ (1o ·o ω) = ω
11		2omomeqom 44013	. . . . . 6 ⊢ (2o ·o ω) = ω
12	10, 11	eleq12i 2856	. . . . 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))

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2143  ∅c0 4287  {cpr 4592  Ord word 6361  Oncon0 6362  (class class class)co 7412  ωcom 7863  1_oc1o 8447  2_oc2o 8448   ·_o comu 8452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734  ax-inf2 9611

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-oadd 8458  df-omul 8459

This theorem is referenced by:  omnord1  44015