Theorem omnord1ex 43845

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 8442	. . . . 5 ⊢ 1o ∈ V
2	1	prid2 4721	. . . 4 ⊢ 1o ∈ {∅, 1o}
3		df2o3 8440	. . . 4 ⊢ 2o = {∅, 1o}
4	2, 3	eleqtrri 2860	. . 3 ⊢ 1o ∈ 2o
5		ordom 7852	. . . 4 ⊢ Ord ω
6		ordirr 6360	. . . . 5 ⊢ (Ord ω → ¬ ω ∈ ω)
7		omelon 9598	. . . . . . 7 ⊢ ω ∈ On
8		1onn 8605	. . . . . . 7 ⊢ 1o ∈ ω
9		0lt1o 8468	. . . . . . 7 ⊢ ∅ ∈ 1o
10		omabslem 8615	. . . . . . 7 ⊢ ((ω ∈ On ∧ 1o ∈ ω ∧ ∅ ∈ 1o) → (1o ·o ω) = ω)
11	7, 8, 9, 10	mp3an 1481	. . . . . 6 ⊢ (1o ·o ω) = ω
12		2omomeqom 43844	. . . . . 6 ⊢ (2o ·o ω) = ω
13	11, 12	eleq12i 2854	. . . . 5 ⊢ ((1o ·o ω) ∈ (2o ·o ω) ↔ ω ∈ ω)
14	6, 13	sylnibr 331	. . . 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 266	. 2 ⊢ (1o ∈ 2o ↔ ¬ (1o ·o ω) ∈ (2o ·o ω))
17		xor3 384	. 2 ⊢ (¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω)) ↔ (1o ∈ 2o ↔ ¬ (1o ·o ω) ∈ (2o ·o ω)))
18	16, 17	mpbir 233	1 ⊢ ¬ (1o ∈ 2o ↔ (1o ·o ω) ∈ (2o ·o ω))

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1559   ∈ wcel 2141  ∅c0 4285  {cpr 4583  Ord word 6341  Oncon0 6342  (class class class)co 7392  ωcom 7842  1_oc1o 8425  2_oc2o 8426   ·_o comu 8430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389  ax-un 7714  ax-inf2 9593

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-2o 8433  df-oadd 8436  df-omul 8437

This theorem is referenced by:  omnord1  43846