Theorem oenord1 41901

Description: When two ordinals (both at least as large as two) are raised to the same power, ordering of the powers is not equivalent to the ordering of the bases. Remark 3.26 of [Schloeder] p. 11. (Contributed by RP, 4-Feb-2025.)

Assertion

Ref	Expression
oenord1	⊢ ∃𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))

Distinct variable group:   𝑎,𝑏,𝑐

Proof of Theorem oenord1

Step	Hyp	Ref	Expression
1		oenord1ex 41900	. 2 ⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))
2		2on 8464	. . . 4 ⊢ 2o ∈ On
3		1oex 8460	. . . . . 6 ⊢ 1o ∈ V
4	3	prid2 4761	. . . . 5 ⊢ 1o ∈ {∅, 1o}
5		df2o3 8458	. . . . 5 ⊢ 2o = {∅, 1o}
6	4, 5	eleqtrri 2832	. . . 4 ⊢ 1o ∈ 2o
7		ondif2 8486	. . . 4 ⊢ (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
8	2, 6, 7	mpbir2an 709	. . 3 ⊢ 2o ∈ (On ∖ 2o)
9		3on 8468	. . . . 5 ⊢ 3o ∈ On
10	3	tpid2 4768	. . . . . 6 ⊢ 1o ∈ {∅, 1o, 2o}
11		df3o2 41898	. . . . . 6 ⊢ 3o = {∅, 1o, 2o}
12	10, 11	eleqtrri 2832	. . . . 5 ⊢ 1o ∈ 3o
13		ondif2 8486	. . . . 5 ⊢ (3o ∈ (On ∖ 2o) ↔ (3o ∈ On ∧ 1o ∈ 3o))
14	9, 12, 13	mpbir2an 709	. . . 4 ⊢ 3o ∈ (On ∖ 2o)
15		omelon 9625	. . . . . 6 ⊢ ω ∈ On
16		peano1 7863	. . . . . 6 ⊢ ∅ ∈ ω
17		ondif1 8485	. . . . . 6 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
18	15, 16, 17	mpbir2an 709	. . . . 5 ⊢ ω ∈ (On ∖ 1o)
19		oveq2 7402	. . . . . . . . 9 ⊢ (𝑐 = ω → (2o ↑o 𝑐) = (2o ↑o ω))
20		oveq2 7402	. . . . . . . . 9 ⊢ (𝑐 = ω → (3o ↑o 𝑐) = (3o ↑o ω))
21	19, 20	eleq12d 2827	. . . . . . . 8 ⊢ (𝑐 = ω → ((2o ↑o 𝑐) ∈ (3o ↑o 𝑐) ↔ (2o ↑o ω) ∈ (3o ↑o ω)))
22	21	bibi2d 342	. . . . . . 7 ⊢ (𝑐 = ω → ((2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)) ↔ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))))
23	22	notbid 317	. . . . . 6 ⊢ (𝑐 = ω → (¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)) ↔ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))))
24	23	rspcev 3610	. . . . 5 ⊢ ((ω ∈ (On ∖ 1o) ∧ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))) → ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)))
25	18, 24	mpan 688	. . . 4 ⊢ (¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) → ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)))
26		eleq2 2822	. . . . . . . 8 ⊢ (𝑏 = 3o → (2o ∈ 𝑏 ↔ 2o ∈ 3o))
27		oveq1 7401	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝑏 ↑o 𝑐) = (3o ↑o 𝑐))
28	27	eleq2d 2819	. . . . . . . 8 ⊢ (𝑏 = 3o → ((2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐) ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)))
29	26, 28	bibi12d 345	. . . . . . 7 ⊢ (𝑏 = 3o → ((2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐))))
30	29	notbid 317	. . . . . 6 ⊢ (𝑏 = 3o → (¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐))))
31	30	rexbidv 3178	. . . . 5 ⊢ (𝑏 = 3o → (∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐))))
32	31	rspcev 3610	. . . 4 ⊢ ((3o ∈ (On ∖ 2o) ∧ ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐))) → ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)))
33	14, 25, 32	sylancr 587	. . 3 ⊢ (¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) → ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)))
34		eleq1 2821	. . . . . . 7 ⊢ (𝑎 = 2o → (𝑎 ∈ 𝑏 ↔ 2o ∈ 𝑏))
35		oveq1 7401	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑐) = (2o ↑o 𝑐))
36	35	eleq1d 2818	. . . . . . 7 ⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐) ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)))
37	34, 36	bibi12d 345	. . . . . 6 ⊢ (𝑎 = 2o → ((𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))))
38	37	notbid 317	. . . . 5 ⊢ (𝑎 = 2o → (¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))))
39	38	2rexbidv 3219	. . . 4 ⊢ (𝑎 = 2o → (∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))))
40	39	rspcev 3610	. . 3 ⊢ ((2o ∈ (On ∖ 2o) ∧ ∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))) → ∃𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)))
41	8, 33, 40	sylancr 587	. 2 ⊢ (¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) → ∃𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)))
42	1, 41	ax-mp 5	1 ⊢ ∃𝑎 ∈ (On ∖ 2o)∃𝑏 ∈ (On ∖ 2o)∃𝑐 ∈ (On ∖ 1o) ¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1541   ∈ wcel 2106  ∃wrex 3070   ∖ cdif 3942  ∅c0 4319  {cpr 4625  {ctp 4627  Oncon0 6354  (class class class)co 7394  ωcom 7839  1_oc1o 8443  2_oc2o 8444  3_oc3o 8445   ↑_o coe 8449

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5421  ax-un 7709  ax-inf2 9620

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7840  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-1o 8450  df-2o 8451  df-3o 8452  df-oadd 8454  df-omul 8455  df-oexp 8456

This theorem is referenced by: (None)