Theorem oenass 43777

Description: Ordinal exponentiation is not associative. Remark 4.6 of [Schloeder] p. 14. (Contributed by RP, 30-Jan-2025.)

Assertion

Ref	Expression
oenass	⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐)

Distinct variable group:   𝑎,𝑏,𝑐

Proof of Theorem oenass

Step	Hyp	Ref	Expression
1		oenassex 43776	. 2 ⊢ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)
2		2on 8412	. . 3 ⊢ 2o ∈ On
3		0elon 6368	. . . . 5 ⊢ ∅ ∈ On
4		oveq2 7367	. . . . . . . . 9 ⊢ (𝑐 = ∅ → (2o ↑o 𝑐) = (2o ↑o ∅))
5	4	oveq2d 7375	. . . . . . . 8 ⊢ (𝑐 = ∅ → (2o ↑o (2o ↑o 𝑐)) = (2o ↑o (2o ↑o ∅)))
6		oveq2 7367	. . . . . . . 8 ⊢ (𝑐 = ∅ → ((2o ↑o 2o) ↑o 𝑐) = ((2o ↑o 2o) ↑o ∅))
7	5, 6	eqeq12d 2757	. . . . . . 7 ⊢ (𝑐 = ∅ → ((2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐) ↔ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)))
8	7	notbid 320	. . . . . 6 ⊢ (𝑐 = ∅ → (¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐) ↔ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)))
9	8	rspcev 3561	. . . . 5 ⊢ ((∅ ∈ On ∧ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)) → ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐))
10	3, 9	mpan 697	. . . 4 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) → ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐))
11		oveq1 7366	. . . . . . . . 9 ⊢ (𝑏 = 2o → (𝑏 ↑o 𝑐) = (2o ↑o 𝑐))
12	11	oveq2d 7375	. . . . . . . 8 ⊢ (𝑏 = 2o → (2o ↑o (𝑏 ↑o 𝑐)) = (2o ↑o (2o ↑o 𝑐)))
13		oveq2 7367	. . . . . . . . 9 ⊢ (𝑏 = 2o → (2o ↑o 𝑏) = (2o ↑o 2o))
14	13	oveq1d 7374	. . . . . . . 8 ⊢ (𝑏 = 2o → ((2o ↑o 𝑏) ↑o 𝑐) = ((2o ↑o 2o) ↑o 𝑐))
15	12, 14	eqeq12d 2757	. . . . . . 7 ⊢ (𝑏 = 2o → ((2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐) ↔ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐)))
16	15	notbid 320	. . . . . 6 ⊢ (𝑏 = 2o → (¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐) ↔ ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐)))
17	16	rexbidv 3165	. . . . 5 ⊢ (𝑏 = 2o → (∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐) ↔ ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐)))
18	17	rspcev 3561	. . . 4 ⊢ ((2o ∈ On ∧ ∃𝑐 ∈ On ¬ (2o ↑o (2o ↑o 𝑐)) = ((2o ↑o 2o) ↑o 𝑐)) → ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐))
19	2, 10, 18	sylancr 594	. . 3 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) → ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐))
20		oveq1 7366	. . . . . . . 8 ⊢ (𝑎 = 2o → (𝑎 ↑o (𝑏 ↑o 𝑐)) = (2o ↑o (𝑏 ↑o 𝑐)))
21		oveq1 7366	. . . . . . . . 9 ⊢ (𝑎 = 2o → (𝑎 ↑o 𝑏) = (2o ↑o 𝑏))
22	21	oveq1d 7374	. . . . . . . 8 ⊢ (𝑎 = 2o → ((𝑎 ↑o 𝑏) ↑o 𝑐) = ((2o ↑o 𝑏) ↑o 𝑐))
23	20, 22	eqeq12d 2757	. . . . . . 7 ⊢ (𝑎 = 2o → ((𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐) ↔ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)))
24	23	notbid 320	. . . . . 6 ⊢ (𝑎 = 2o → (¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐) ↔ ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)))
25	24	rexbidv 3165	. . . . 5 ⊢ (𝑎 = 2o → (∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐) ↔ ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)))
26	25	rexbidv 3165	. . . 4 ⊢ (𝑎 = 2o → (∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐) ↔ ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)))
27	26	rspcev 3561	. . 3 ⊢ ((2o ∈ On ∧ ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (2o ↑o (𝑏 ↑o 𝑐)) = ((2o ↑o 𝑏) ↑o 𝑐)) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐))
28	2, 19, 27	sylancr 594	. 2 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) → ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐))
29	1, 28	ax-mp 5	1 ⊢ ∃𝑎 ∈ On ∃𝑏 ∈ On ∃𝑐 ∈ On ¬ (𝑎 ↑o (𝑏 ↑o 𝑐)) = ((𝑎 ↑o 𝑏) ↑o 𝑐)

Syntax hints:  ¬ wn 3   = wceq 1548   ∈ wcel 2121  ∃wrex 3065  ∅c0 4263  Oncon0 6313  (class class class)co 7359  2_oc2o 8393   ↑_o coe 8398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7681  ax-reg 9501

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-1o 8399  df-2o 8400  df-oadd 8403  df-omul 8404  df-oexp 8405

This theorem is referenced by: (None)