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Theorem oenassex 43336
Description: Ordinal two raised to two to the zeroth power is not the same as two squared then raised to the zeroth power. (Contributed by RP, 30-Jan-2025.)
Assertion
Ref Expression
oenassex ¬ (2oo (2oo ∅)) = ((2oo 2o) ↑o ∅)

Proof of Theorem oenassex
StepHypRef Expression
1 1oex 8517 . . . 4 1o ∈ V
21prid2 4762 . . 3 1o ∈ {∅, 1o}
3 df2o3 8515 . . 3 2o = {∅, 1o}
42, 3eleqtrri 2839 . 2 1o ∈ 2o
5 elneq 9639 . . 3 (1o ∈ 2o → 1o ≠ 2o)
6 df-ne 2940 . . . 4 (2o ≠ 1o ↔ ¬ 2o = 1o)
7 necom 2993 . . . 4 (1o ≠ 2o ↔ 2o ≠ 1o)
8 2on 8521 . . . . . . . . 9 2o ∈ On
9 oe0 8561 . . . . . . . . 9 (2o ∈ On → (2oo ∅) = 1o)
108, 9ax-mp 5 . . . . . . . 8 (2oo ∅) = 1o
1110oveq2i 7443 . . . . . . 7 (2oo (2oo ∅)) = (2oo 1o)
12 oe1 8583 . . . . . . . 8 (2o ∈ On → (2oo 1o) = 2o)
138, 12ax-mp 5 . . . . . . 7 (2oo 1o) = 2o
1411, 13eqtri 2764 . . . . . 6 (2oo (2oo ∅)) = 2o
158, 8pm3.2i 470 . . . . . . 7 (2o ∈ On ∧ 2o ∈ On)
16 oecl 8576 . . . . . . 7 ((2o ∈ On ∧ 2o ∈ On) → (2oo 2o) ∈ On)
17 oe0 8561 . . . . . . 7 ((2oo 2o) ∈ On → ((2oo 2o) ↑o ∅) = 1o)
1815, 16, 17mp2b 10 . . . . . 6 ((2oo 2o) ↑o ∅) = 1o
1914, 18eqeq12i 2754 . . . . 5 ((2oo (2oo ∅)) = ((2oo 2o) ↑o ∅) ↔ 2o = 1o)
2019notbii 320 . . . 4 (¬ (2oo (2oo ∅)) = ((2oo 2o) ↑o ∅) ↔ ¬ 2o = 1o)
216, 7, 203bitr4i 303 . . 3 (1o ≠ 2o ↔ ¬ (2oo (2oo ∅)) = ((2oo 2o) ↑o ∅))
225, 21sylib 218 . 2 (1o ∈ 2o → ¬ (2oo (2oo ∅)) = ((2oo 2o) ↑o ∅))
234, 22ax-mp 5 1 ¬ (2oo (2oo ∅)) = ((2oo 2o) ↑o ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1539  wcel 2107  wne 2939  c0 4332  {cpr 4627  Oncon0 6383  (class class class)co 7432  1oc1o 8500  2oc2o 8501  o coe 8506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756  ax-reg 9633
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-omul 8512  df-oexp 8513
This theorem is referenced by:  oenass  43337
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