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Mirrors > Home > NFE Home > Th. List > tfinprop | GIF version |
Description: Properties of the finite T operator for a non-empty natural. Theorem X.1.28 of [Rosser] p. 528. (Contributed by SF, 22-Jan-2015.) |
Ref | Expression |
---|---|
tfinprop | ⊢ ((M ∈ Nn ∧ M ≠ ∅) → ( Tfin M ∈ Nn ∧ ∃a ∈ M ℘1a ∈ Tfin M)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tfin 4443 | . . 3 ⊢ Tfin M = if(M = ∅, ∅, (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n))) | |
2 | df-ne 2518 | . . . . . 6 ⊢ (M ≠ ∅ ↔ ¬ M = ∅) | |
3 | iffalse 3669 | . . . . . 6 ⊢ (¬ M = ∅ → if(M = ∅, ∅, (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n))) = (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n))) | |
4 | 2, 3 | sylbi 187 | . . . . 5 ⊢ (M ≠ ∅ → if(M = ∅, ∅, (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n))) = (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n))) |
5 | 4 | adantl 452 | . . . 4 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → if(M = ∅, ∅, (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n))) = (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n))) |
6 | nnpw1ex 4484 | . . . . 5 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → ∃!n ∈ Nn ∃a ∈ M ℘1a ∈ n) | |
7 | reiotacl 4364 | . . . . 5 ⊢ (∃!n ∈ Nn ∃a ∈ M ℘1a ∈ n → (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n)) ∈ Nn ) | |
8 | 6, 7 | syl 15 | . . . 4 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n)) ∈ Nn ) |
9 | 5, 8 | eqeltrd 2427 | . . 3 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → if(M = ∅, ∅, (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n))) ∈ Nn ) |
10 | 1, 9 | syl5eqel 2437 | . 2 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → Tfin M ∈ Nn ) |
11 | 1, 5 | syl5req 2398 | . . 3 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n)) = Tfin M) |
12 | 10, 6 | jca 518 | . . . 4 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → ( Tfin M ∈ Nn ∧ ∃!n ∈ Nn ∃a ∈ M ℘1a ∈ n)) |
13 | eleq2 2414 | . . . . . 6 ⊢ (n = Tfin M → (℘1a ∈ n ↔ ℘1a ∈ Tfin M)) | |
14 | 13 | rexbidv 2635 | . . . . 5 ⊢ (n = Tfin M → (∃a ∈ M ℘1a ∈ n ↔ ∃a ∈ M ℘1a ∈ Tfin M)) |
15 | 14 | reiota2 4368 | . . . 4 ⊢ (( Tfin M ∈ Nn ∧ ∃!n ∈ Nn ∃a ∈ M ℘1a ∈ n) → (∃a ∈ M ℘1a ∈ Tfin M ↔ (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n)) = Tfin M)) |
16 | 12, 15 | syl 15 | . . 3 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → (∃a ∈ M ℘1a ∈ Tfin M ↔ (℩n(n ∈ Nn ∧ ∃a ∈ M ℘1a ∈ n)) = Tfin M)) |
17 | 11, 16 | mpbird 223 | . 2 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → ∃a ∈ M ℘1a ∈ Tfin M) |
18 | 10, 17 | jca 518 | 1 ⊢ ((M ∈ Nn ∧ M ≠ ∅) → ( Tfin M ∈ Nn ∧ ∃a ∈ M ℘1a ∈ Tfin M)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∃wrex 2615 ∃!wreu 2616 ∅c0 3550 ifcif 3662 ℘1cpw1 4135 ℩cio 4337 Nn cnnc 4373 Tfin ctfin 4435 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-tfin 4443 |
This theorem is referenced by: tfinnnul 4490 tfincl 4492 tfin11 4493 tfinpw1 4494 tfinltfinlem1 4500 tfinltfin 4501 eventfin 4517 oddtfin 4518 sfinltfin 4535 vfinncvntnn 4548 |
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