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Mirrors > Home > MPE Home > Th. List > Mathboxes > oenassex | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
oenassex | ⊢ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 8515 | . . . 4 ⊢ 1o ∈ V | |
2 | 1 | prid2 4768 | . . 3 ⊢ 1o ∈ {∅, 1o} |
3 | df2o3 8513 | . . 3 ⊢ 2o = {∅, 1o} | |
4 | 2, 3 | eleqtrri 2838 | . 2 ⊢ 1o ∈ 2o |
5 | elneq 9636 | . . 3 ⊢ (1o ∈ 2o → 1o ≠ 2o) | |
6 | df-ne 2939 | . . . 4 ⊢ (2o ≠ 1o ↔ ¬ 2o = 1o) | |
7 | necom 2992 | . . . 4 ⊢ (1o ≠ 2o ↔ 2o ≠ 1o) | |
8 | 2on 8519 | . . . . . . . . 9 ⊢ 2o ∈ On | |
9 | oe0 8559 | . . . . . . . . 9 ⊢ (2o ∈ On → (2o ↑o ∅) = 1o) | |
10 | 8, 9 | ax-mp 5 | . . . . . . . 8 ⊢ (2o ↑o ∅) = 1o |
11 | 10 | oveq2i 7442 | . . . . . . 7 ⊢ (2o ↑o (2o ↑o ∅)) = (2o ↑o 1o) |
12 | oe1 8581 | . . . . . . . 8 ⊢ (2o ∈ On → (2o ↑o 1o) = 2o) | |
13 | 8, 12 | ax-mp 5 | . . . . . . 7 ⊢ (2o ↑o 1o) = 2o |
14 | 11, 13 | eqtri 2763 | . . . . . 6 ⊢ (2o ↑o (2o ↑o ∅)) = 2o |
15 | 8, 8 | pm3.2i 470 | . . . . . . 7 ⊢ (2o ∈ On ∧ 2o ∈ On) |
16 | oecl 8574 | . . . . . . 7 ⊢ ((2o ∈ On ∧ 2o ∈ On) → (2o ↑o 2o) ∈ On) | |
17 | oe0 8559 | . . . . . . 7 ⊢ ((2o ↑o 2o) ∈ On → ((2o ↑o 2o) ↑o ∅) = 1o) | |
18 | 15, 16, 17 | mp2b 10 | . . . . . 6 ⊢ ((2o ↑o 2o) ↑o ∅) = 1o |
19 | 14, 18 | eqeq12i 2753 | . . . . 5 ⊢ ((2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) ↔ 2o = 1o) |
20 | 19 | notbii 320 | . . . 4 ⊢ (¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) ↔ ¬ 2o = 1o) |
21 | 6, 7, 20 | 3bitr4i 303 | . . 3 ⊢ (1o ≠ 2o ↔ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)) |
22 | 5, 21 | sylib 218 | . 2 ⊢ (1o ∈ 2o → ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅)) |
23 | 4, 22 | ax-mp 5 | 1 ⊢ ¬ (2o ↑o (2o ↑o ∅)) = ((2o ↑o 2o) ↑o ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∅c0 4339 {cpr 4633 Oncon0 6386 (class class class)co 7431 1oc1o 8498 2oc2o 8499 ↑o coe 8504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 ax-reg 9630 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-oexp 8511 |
This theorem is referenced by: oenass 43309 |
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