Theorem oenord1 44261

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 44260	. 2 ⊢ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))
2		2on 8468	. . . 4 ⊢ 2o ∈ On
3		1oelpr 8465	. . . . 5 ⊢ 1o ∈ {∅, 1o}
4		df2o3 8462	. . . . 5 ⊢ 2o = {∅, 1o}
5	3, 4	eleqtrri 2859	. . . 4 ⊢ 1o ∈ 2o
6		ondif2 8488	. . . 4 ⊢ (2o ∈ (On ∖ 2o) ↔ (2o ∈ On ∧ 1o ∈ 2o))
7	2, 5, 6	mpbir2an 724	. . 3 ⊢ 2o ∈ (On ∖ 2o)
8		3on 8471	. . . . 5 ⊢ 3o ∈ On
9		1oex 8464	. . . . . . 7 ⊢ 1o ∈ V
10	9	tpid2 4730	. . . . . 6 ⊢ 1o ∈ {∅, 1o, 2o}
11		df3o2 44258	. . . . . 6 ⊢ 3o = {∅, 1o, 2o}
12	10, 11	eleqtrri 2859	. . . . 5 ⊢ 1o ∈ 3o
13		ondif2 8488	. . . . 5 ⊢ (3o ∈ (On ∖ 2o) ↔ (3o ∈ On ∧ 1o ∈ 3o))
14	8, 12, 13	mpbir2an 724	. . . 4 ⊢ 3o ∈ (On ∖ 2o)
15		omelon 9625	. . . . . 6 ⊢ ω ∈ On
16		peano1 7883	. . . . . 6 ⊢ ∅ ∈ ω
17		ondif1 8487	. . . . . 6 ⊢ (ω ∈ (On ∖ 1o) ↔ (ω ∈ On ∧ ∅ ∈ ω))
18	15, 16, 17	mpbir2an 724	. . . . 5 ⊢ ω ∈ (On ∖ 1o)
19		oveq2 7416	. . . . . . . . 9 ⊢ (𝑐 = ω → (2o ↑o 𝑐) = (2o ↑o ω))
20		oveq2 7416	. . . . . . . . 9 ⊢ (𝑐 = ω → (3o ↑o 𝑐) = (3o ↑o ω))
21	19, 20	eleq12d 2854	. . . . . . . 8 ⊢ (𝑐 = ω → ((2o ↑o 𝑐) ∈ (3o ↑o 𝑐) ↔ (2o ↑o ω) ∈ (3o ↑o ω)))
22	21	bibi2d 345	. . . . . . 7 ⊢ (𝑐 = ω → ((2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)) ↔ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))))
23	22	notbid 321	. . . . . 6 ⊢ (𝑐 = ω → (¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)) ↔ ¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω))))
24	23	rspcev 3576	. . . . 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 703	. . . 4 ⊢ (¬ (2o ∈ 3o ↔ (2o ↑o ω) ∈ (3o ↑o ω)) → ∃𝑐 ∈ (On ∖ 1o) ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)))
26		eleq2 2849	. . . . . . . 8 ⊢ (𝑏 = 3o → (2o ∈ 𝑏 ↔ 2o ∈ 3o))
27		oveq1 7415	. . . . . . . . 9 ⊢ (𝑏 = 3o → (𝑏 ↑o 𝑐) = (3o ↑o 𝑐))
28	27	eleq2d 2846	. . . . . . . 8 ⊢ (𝑏 = 3o → ((2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐) ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐)))
29	26, 28	bibi12d 348	. . . . . . 7 ⊢ (𝑏 = 3o → ((2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐))))
30	29	notbid 321	. . . . . 6 ⊢ (𝑏 = 3o → (¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ ¬ (2o ∈ 3o ↔ (2o ↑o 𝑐) ∈ (3o ↑o 𝑐))))
31	30	rexbidv 3186	. . . . 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 3576	. . . 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 599	. . 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 2848	. . . . . . 7 ⊢ (𝑎 = 2o → (𝑎 ∈ 𝑏 ↔ 2o ∈ 𝑏))
35		oveq1 7415	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑐) = (2o ↑o 𝑐))
36	35	eleq1d 2845	. . . . . . 7 ⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐) ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)))
37	34, 36	bibi12d 348	. . . . . 6 ⊢ (𝑎 = 2o → ((𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))))
38	37	notbid 321	. . . . 5 ⊢ (𝑎 = 2o → (¬ (𝑎 ∈ 𝑏 ↔ (𝑎 ↑o 𝑐) ∈ (𝑏 ↑o 𝑐)) ↔ ¬ (2o ∈ 𝑏 ↔ (2o ↑o 𝑐) ∈ (𝑏 ↑o 𝑐))))
39	38	2rexbidv 3227	. . . 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 3576	. . 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	7, 33, 40	sylancr 599	. 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))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∃wrex 3086   ∖ cdif 3895  ∅c0 4278  {cpr 4585  {ctp 4587  Oncon0 6351  (class class class)co 7408  ωcom 7860  1_oc1o 8447  2_oc2o 8448  3_oc3o 8449   ↑_o coe 8453

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 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7734  ax-inf2 9620

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 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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-3o 8456  df-oadd 8458  df-omul 8459  df-oexp 8460

This theorem is used by: (None)