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Theorem oenassex 42523
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 8471 . . . 4 1o ∈ V
21prid2 4759 . . 3 1o ∈ {∅, 1o}
3 df2o3 8469 . . 3 2o = {∅, 1o}
42, 3eleqtrri 2824 . 2 1o ∈ 2o
5 elneq 9588 . . 3 (1o ∈ 2o → 1o ≠ 2o)
6 df-ne 2933 . . . 4 (2o ≠ 1o ↔ ¬ 2o = 1o)
7 necom 2986 . . . 4 (1o ≠ 2o ↔ 2o ≠ 1o)
8 2on 8475 . . . . . . . . 9 2o ∈ On
9 oe0 8517 . . . . . . . . 9 (2o ∈ On → (2oo ∅) = 1o)
108, 9ax-mp 5 . . . . . . . 8 (2oo ∅) = 1o
1110oveq2i 7412 . . . . . . 7 (2oo (2oo ∅)) = (2oo 1o)
12 oe1 8539 . . . . . . . 8 (2o ∈ On → (2oo 1o) = 2o)
138, 12ax-mp 5 . . . . . . 7 (2oo 1o) = 2o
1411, 13eqtri 2752 . . . . . 6 (2oo (2oo ∅)) = 2o
158, 8pm3.2i 470 . . . . . . 7 (2o ∈ On ∧ 2o ∈ On)
16 oecl 8532 . . . . . . 7 ((2o ∈ On ∧ 2o ∈ On) → (2oo 2o) ∈ On)
17 oe0 8517 . . . . . . 7 ((2oo 2o) ∈ On → ((2oo 2o) ↑o ∅) = 1o)
1815, 16, 17mp2b 10 . . . . . 6 ((2oo 2o) ↑o ∅) = 1o
1914, 18eqeq12i 2742 . . . . 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 217 . 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 1533  wcel 2098  wne 2932  c0 4314  {cpr 4622  Oncon0 6354  (class class class)co 7401  1oc1o 8454  2oc2o 8455  o coe 8460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718  ax-reg 9582
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  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 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-2o 8462  df-oadd 8465  df-omul 8466  df-oexp 8467
This theorem is referenced by:  oenass  42524
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