Theorem omnord1ex 44400

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 8491	. . . 4 ⊢ 1o ∈ {∅, 1o}
2		df2o3 8488	. . . 4 ⊢ 2o = {∅, 1o}
3	1, 2	eleqtrri 2859	. . 3 ⊢ 1o ∈ 2o
4		ordom 7888	. . . 4 ⊢ Ord ω
5		ordirr 6380	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
6		omelon 9651	. . . . . . 7 ⊢ ω ∈ On
7		1onn 8653	. . . . . . 7 ⊢ 1o ∈ ω
8		0lt1o 8516	. . . . . . 7 ⊢ ∅ ∈ 1o
9		omabslem 8663	. . . . . . 7 ⊢ ((ω ∈ On ∧ 1o ∈ ω ∧ ∅ ∈ 1o) → (1o ·o ω) = ω)
10	6, 7, 8, 9	mp3an 1490	. . . . . 6 ⊢ (1o ·o ω) = ω
11		2omomeqom 44399	. . . . . 6 ⊢ (2o ·o ω) = ω
12	10, 11	eleq12i 2853	. . . . 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 1569   ∈ wcel 2144  ∅c0 4278  {cpr 4586  Ord word 6361  Oncon0 6362  (class class class)co 7421  ωcom 7878  1_oc1o 8473  2_oc2o 8474   ·_o comu 8478

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752  ax-inf2 9646

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-om 7879  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-1o 8480  df-2o 8481  df-oadd 8484  df-omul 8485

This theorem is used by:  omnord1  44401