Theorem omnord1ex 44153

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 8470	. . . 4 ⊢ 1o ∈ {∅, 1o}
2		df2o3 8467	. . . 4 ⊢ 2o = {∅, 1o}
3	1, 2	eleqtrri 2861	. . 3 ⊢ 1o ∈ 2o
4		ordom 7876	. . . 4 ⊢ Ord ω
5		ordirr 6379	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
6		omelon 9629	. . . . . . 7 ⊢ ω ∈ On
7		1onn 8632	. . . . . . 7 ⊢ 1o ∈ ω
8		0lt1o 8495	. . . . . . 7 ⊢ ∅ ∈ 1o
9		omabslem 8642	. . . . . . 7 ⊢ ((ω ∈ On ∧ 1o ∈ ω ∧ ∅ ∈ 1o) → (1o ·o ω) = ω)
10	6, 7, 8, 9	mp3an 1490	. . . . . 6 ⊢ (1o ·o ω) = ω
11		2omomeqom 44152	. . . . . 6 ⊢ (2o ·o ω) = ω
12	10, 11	eleq12i 2855	. . . . 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 2145  ∅c0 4282  {cpr 4589  Ord word 6360  Oncon0 6361  (class class class)co 7417  ωcom 7866  1_oc1o 8452  2_oc2o 8453   ·_o comu 8457

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740  ax-inf2 9624

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-oadd 8463  df-omul 8464

This theorem is used by:  omnord1  44154