Theorem omnord1ex 43750

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		1oex 8408	. . . . 5 ⊢ 1o ∈ V
2	1	prid2 4708	. . . 4 ⊢ 1o ∈ {∅, 1o}
3		df2o3 8406	. . . 4 ⊢ 2o = {∅, 1o}
4	2, 3	eleqtrri 2836	. . 3 ⊢ 1o ∈ 2o
5		ordom 7820	. . . 4 ⊢ Ord ω
6		ordirr 6335	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
7		omelon 9558	. . . . . . 7 ⊢ ω ∈ On
8		1onn 8569	. . . . . . 7 ⊢ 1o ∈ ω
9		0lt1o 8432	. . . . . . 7 ⊢ ∅ ∈ 1o
10		omabslem 8579	. . . . . . 7 ⊢ ((ω ∈ On ∧ 1o ∈ ω ∧ ∅ ∈ 1o) → (1o ·o ω) = ω)
11	7, 8, 9, 10	mp3an 1464	. . . . . 6 ⊢ (1o ·o ω) = ω
12		2omomeqom 43749	. . . . . 6 ⊢ (2o ·o ω) = ω
13	11, 12	eleq12i 2830	. . . . 5 ⊢ ((1o ·o ω) ∈ (2o ·o ω) ↔ ω ∈ ω)
14	6, 13	sylnibr 329	. . . 4 ⊢ (Ord ω → ¬ (1o ·o ω) ∈ (2o ·o ω))
15	5, 14	ax-mp 5	. . 3 ⊢ ¬ (1o ·o ω) ∈ (2o ·o ω)
16	4, 15	2th 264	. 2 ⊢ (1o ∈ 2o ↔ ¬ (1o ·o ω) ∈ (2o ·o ω))
17		xor3 382	. 2 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) ↔ (1o ∈ 2o ↔ ¬ (1o ·o ω) ∈ (2o ·o ω)))
18	16, 17	mpbir 231	1 ⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∅c0 4274  {cpr 4570  Ord word 6316  Oncon0 6317  (class class class)co 7360  ωcom 7810  1_oc1o 8391  2_oc2o 8392   ·_o comu 8396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5370  ax-un 7682  ax-inf2 9553

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-1o 8398  df-2o 8399  df-oadd 8402  df-omul 8403

This theorem is referenced by:  omnord1  43751