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Theorem nnaordex 6519
Description: Equivalence for ordering. Compare Exercise 23 of [Enderton] p. 88. (Contributed by NM, 5-Dec-1995.) (Revised by Mario Carneiro, 15-Nov-2014.)
Assertion
Ref Expression
nnaordex ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nnaordex
Dummy variables 𝑏 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2239 . . . . . 6 (𝑏 = 𝐵 → (𝐴𝑏𝐴𝐵))
2 eqeq2 2185 . . . . . . . 8 (𝑏 = 𝐵 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = 𝐵))
32anbi2d 464 . . . . . . 7 (𝑏 = 𝐵 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
43rexbidv 2476 . . . . . 6 (𝑏 = 𝐵 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
51, 4imbi12d 234 . . . . 5 (𝑏 = 𝐵 → ((𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
65imbi2d 230 . . . 4 (𝑏 = 𝐵 → ((𝐴 ∈ ω → (𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏))) ↔ (𝐴 ∈ ω → (𝐴𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))))
7 eleq2 2239 . . . . . 6 (𝑏 = ∅ → (𝐴𝑏𝐴 ∈ ∅))
8 eqeq2 2185 . . . . . . . 8 (𝑏 = ∅ → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = ∅))
98anbi2d 464 . . . . . . 7 (𝑏 = ∅ → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
109rexbidv 2476 . . . . . 6 (𝑏 = ∅ → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
117, 10imbi12d 234 . . . . 5 (𝑏 = ∅ → ((𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅))))
12 eleq2 2239 . . . . . 6 (𝑏 = 𝑦 → (𝐴𝑏𝐴𝑦))
13 eqeq2 2185 . . . . . . . 8 (𝑏 = 𝑦 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = 𝑦))
1413anbi2d 464 . . . . . . 7 (𝑏 = 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)))
1514rexbidv 2476 . . . . . 6 (𝑏 = 𝑦 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)))
1612, 15imbi12d 234 . . . . 5 (𝑏 = 𝑦 → ((𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦))))
17 eleq2 2239 . . . . . 6 (𝑏 = suc 𝑦 → (𝐴𝑏𝐴 ∈ suc 𝑦))
18 eqeq2 2185 . . . . . . . 8 (𝑏 = suc 𝑦 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = suc 𝑦))
1918anbi2d 464 . . . . . . 7 (𝑏 = suc 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
2019rexbidv 2476 . . . . . 6 (𝑏 = suc 𝑦 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
2117, 20imbi12d 234 . . . . 5 (𝑏 = suc 𝑦 → ((𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴 ∈ suc 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))))
22 noel 3424 . . . . . . 7 ¬ 𝐴 ∈ ∅
2322pm2.21i 646 . . . . . 6 (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅))
2423a1i 9 . . . . 5 (𝐴 ∈ ω → (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
25 elsuci 4397 . . . . . . 7 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
26 simpr 110 . . . . . . . . 9 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦))) → (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)))
27 peano2 4588 . . . . . . . . . . . . . . 15 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
2827ad2antlr 489 . . . . . . . . . . . . . 14 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → suc 𝑥 ∈ ω)
29 elelsuc 4403 . . . . . . . . . . . . . . . . 17 (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥)
3029a1i 9 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥))
31 nnasuc 6467 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
32 suceq 4396 . . . . . . . . . . . . . . . . . 18 ((𝐴 +o 𝑥) = 𝑦 → suc (𝐴 +o 𝑥) = suc 𝑦)
3331, 32sylan9eq 2228 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (𝐴 +o 𝑥) = 𝑦) → (𝐴 +o suc 𝑥) = suc 𝑦)
3433ex 115 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +o 𝑥) = 𝑦 → (𝐴 +o suc 𝑥) = suc 𝑦))
3530, 34anim12d 335 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦)))
3635imp 124 . . . . . . . . . . . . . 14 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦))
37 eleq2 2239 . . . . . . . . . . . . . . . 16 (𝑧 = suc 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ suc 𝑥))
38 oveq2 5873 . . . . . . . . . . . . . . . . 17 (𝑧 = suc 𝑥 → (𝐴 +o 𝑧) = (𝐴 +o suc 𝑥))
3938eqeq1d 2184 . . . . . . . . . . . . . . . 16 (𝑧 = suc 𝑥 → ((𝐴 +o 𝑧) = suc 𝑦 ↔ (𝐴 +o suc 𝑥) = suc 𝑦))
4037, 39anbi12d 473 . . . . . . . . . . . . . . 15 (𝑧 = suc 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦)))
4140rspcev 2839 . . . . . . . . . . . . . 14 ((suc 𝑥 ∈ ω ∧ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦)) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦))
4228, 36, 41syl2anc 411 . . . . . . . . . . . . 13 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦))
4342ex 115 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦)))
4443rexlimdva 2592 . . . . . . . . . . 11 (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦)))
45 eleq2 2239 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ 𝑥))
46 oveq2 5873 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝐴 +o 𝑧) = (𝐴 +o 𝑥))
4746eqeq1d 2184 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ((𝐴 +o 𝑧) = suc 𝑦 ↔ (𝐴 +o 𝑥) = suc 𝑦))
4845, 47anbi12d 473 . . . . . . . . . . . 12 (𝑧 = 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
4948cbvrexv 2702 . . . . . . . . . . 11 (∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
5044, 49syl6ib 161 . . . . . . . . . 10 (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
5150ad2antlr 489 . . . . . . . . 9 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦))) → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
5226, 51syld 45 . . . . . . . 8 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦))) → (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
53 0lt1o 6431 . . . . . . . . . . . 12 ∅ ∈ 1o
5453a1i 9 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∅ ∈ 1o)
55 nnon 4603 . . . . . . . . . . . . 13 (𝐴 ∈ ω → 𝐴 ∈ On)
56 oa1suc 6458 . . . . . . . . . . . . 13 (𝐴 ∈ On → (𝐴 +o 1o) = suc 𝐴)
5755, 56syl 14 . . . . . . . . . . . 12 (𝐴 ∈ ω → (𝐴 +o 1o) = suc 𝐴)
58 suceq 4396 . . . . . . . . . . . 12 (𝐴 = 𝑦 → suc 𝐴 = suc 𝑦)
5957, 58sylan9eq 2228 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → (𝐴 +o 1o) = suc 𝑦)
60 1onn 6511 . . . . . . . . . . . 12 1o ∈ ω
61 eleq2 2239 . . . . . . . . . . . . . 14 (𝑥 = 1o → (∅ ∈ 𝑥 ↔ ∅ ∈ 1o))
62 oveq2 5873 . . . . . . . . . . . . . . 15 (𝑥 = 1o → (𝐴 +o 𝑥) = (𝐴 +o 1o))
6362eqeq1d 2184 . . . . . . . . . . . . . 14 (𝑥 = 1o → ((𝐴 +o 𝑥) = suc 𝑦 ↔ (𝐴 +o 1o) = suc 𝑦))
6461, 63anbi12d 473 . . . . . . . . . . . . 13 (𝑥 = 1o → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦) ↔ (∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦)))
6564rspcev 2839 . . . . . . . . . . . 12 ((1o ∈ ω ∧ (∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦)) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
6660, 65mpan 424 . . . . . . . . . . 11 ((∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
6754, 59, 66syl2anc 411 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
6867ex 115 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 = 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
6968ad2antlr 489 . . . . . . . 8 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦))) → (𝐴 = 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
7052, 69jaod 717 . . . . . . 7 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦))) → ((𝐴𝑦𝐴 = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
7125, 70syl5 32 . . . . . 6 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦))) → (𝐴 ∈ suc 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
7271exp31 364 . . . . 5 (𝑦 ∈ ω → (𝐴 ∈ ω → ((𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → (𝐴 ∈ suc 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))))
7311, 16, 21, 24, 72finds2 4594 . . . 4 (𝑏 ∈ ω → (𝐴 ∈ ω → (𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏))))
746, 73vtoclga 2801 . . 3 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
7574impcom 125 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
76 peano1 4587 . . . . . . . . 9 ∅ ∈ ω
77 nnaord 6500 . . . . . . . . 9 ((∅ ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
7876, 77mp3an1 1324 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
7978ancoms 268 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
80 nna0 6465 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
8180adantr 276 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o ∅) = 𝐴)
8281eleq1d 2244 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +o ∅) ∈ (𝐴 +o 𝑥) ↔ 𝐴 ∈ (𝐴 +o 𝑥)))
8379, 82bitrd 188 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥𝐴 ∈ (𝐴 +o 𝑥)))
8483anbi1d 465 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) ↔ (𝐴 ∈ (𝐴 +o 𝑥) ∧ (𝐴 +o 𝑥) = 𝐵)))
85 eleq2 2239 . . . . . 6 ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +o 𝑥) ↔ 𝐴𝐵))
8685biimpac 298 . . . . 5 ((𝐴 ∈ (𝐴 +o 𝑥) ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴𝐵)
8784, 86syl6bi 163 . . . 4 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴𝐵))
8887rexlimdva 2592 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴𝐵))
8988adantr 276 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴𝐵))
9075, 89impbid 129 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708   = wceq 1353  wcel 2146  wrex 2454  c0 3420  Oncon0 4357  suc csuc 4359  ωcom 4583  (class class class)co 5865  1oc1o 6400   +o coa 6404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-iinf 4581
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-tr 4097  df-id 4287  df-iord 4360  df-on 4362  df-suc 4365  df-iom 4584  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-1st 6131  df-2nd 6132  df-recs 6296  df-irdg 6361  df-1o 6407  df-oadd 6411
This theorem is referenced by:  nnawordex  6520  ltexpi  7311
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