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Theorem nnaordex 6391
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 2181 . . . . . 6 (𝑏 = 𝐵 → (𝐴𝑏𝐴𝐵))
2 eqeq2 2127 . . . . . . . 8 (𝑏 = 𝐵 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = 𝐵))
32anbi2d 459 . . . . . . 7 (𝑏 = 𝐵 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
43rexbidv 2415 . . . . . 6 (𝑏 = 𝐵 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
51, 4imbi12d 233 . . . . 5 (𝑏 = 𝐵 → ((𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
65imbi2d 229 . . . 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 2181 . . . . . 6 (𝑏 = ∅ → (𝐴𝑏𝐴 ∈ ∅))
8 eqeq2 2127 . . . . . . . 8 (𝑏 = ∅ → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = ∅))
98anbi2d 459 . . . . . . 7 (𝑏 = ∅ → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
109rexbidv 2415 . . . . . 6 (𝑏 = ∅ → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
117, 10imbi12d 233 . . . . 5 (𝑏 = ∅ → ((𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅))))
12 eleq2 2181 . . . . . 6 (𝑏 = 𝑦 → (𝐴𝑏𝐴𝑦))
13 eqeq2 2127 . . . . . . . 8 (𝑏 = 𝑦 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = 𝑦))
1413anbi2d 459 . . . . . . 7 (𝑏 = 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)))
1514rexbidv 2415 . . . . . 6 (𝑏 = 𝑦 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)))
1612, 15imbi12d 233 . . . . 5 (𝑏 = 𝑦 → ((𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏)) ↔ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦))))
17 eleq2 2181 . . . . . 6 (𝑏 = suc 𝑦 → (𝐴𝑏𝐴 ∈ suc 𝑦))
18 eqeq2 2127 . . . . . . . 8 (𝑏 = suc 𝑦 → ((𝐴 +o 𝑥) = 𝑏 ↔ (𝐴 +o 𝑥) = suc 𝑦))
1918anbi2d 459 . . . . . . 7 (𝑏 = suc 𝑦 → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
2019rexbidv 2415 . . . . . 6 (𝑏 = suc 𝑦 → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
2117, 20imbi12d 233 . . . . 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 3337 . . . . . . 7 ¬ 𝐴 ∈ ∅
2322pm2.21i 620 . . . . . 6 (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅))
2423a1i 9 . . . . 5 (𝐴 ∈ ω → (𝐴 ∈ ∅ → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = ∅)))
25 elsuci 4295 . . . . . . 7 (𝐴 ∈ suc 𝑦 → (𝐴𝑦𝐴 = 𝑦))
26 simpr 109 . . . . . . . . 9 (((𝑦 ∈ ω ∧ 𝐴 ∈ ω) ∧ (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦))) → (𝐴𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)))
27 peano2 4479 . . . . . . . . . . . . . . 15 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
2827ad2antlr 480 . . . . . . . . . . . . . 14 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → suc 𝑥 ∈ ω)
29 elelsuc 4301 . . . . . . . . . . . . . . . . 17 (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥)
3029a1i 9 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 → ∅ ∈ suc 𝑥))
31 nnasuc 6340 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o suc 𝑥) = suc (𝐴 +o 𝑥))
32 suceq 4294 . . . . . . . . . . . . . . . . . 18 ((𝐴 +o 𝑥) = 𝑦 → suc (𝐴 +o 𝑥) = suc 𝑦)
3331, 32sylan9eq 2170 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (𝐴 +o 𝑥) = 𝑦) → (𝐴 +o suc 𝑥) = suc 𝑦)
3433ex 114 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +o 𝑥) = 𝑦 → (𝐴 +o suc 𝑥) = suc 𝑦))
3530, 34anim12d 333 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦)))
3635imp 123 . . . . . . . . . . . . . 14 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦))
37 eleq2 2181 . . . . . . . . . . . . . . . 16 (𝑧 = suc 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ suc 𝑥))
38 oveq2 5750 . . . . . . . . . . . . . . . . 17 (𝑧 = suc 𝑥 → (𝐴 +o 𝑧) = (𝐴 +o suc 𝑥))
3938eqeq1d 2126 . . . . . . . . . . . . . . . 16 (𝑧 = suc 𝑥 → ((𝐴 +o 𝑧) = suc 𝑦 ↔ (𝐴 +o suc 𝑥) = suc 𝑦))
4037, 39anbi12d 464 . . . . . . . . . . . . . . 15 (𝑧 = suc 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦)))
4140rspcev 2763 . . . . . . . . . . . . . 14 ((suc 𝑥 ∈ ω ∧ (∅ ∈ suc 𝑥 ∧ (𝐴 +o suc 𝑥) = suc 𝑦)) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦))
4228, 36, 41syl2anc 408 . . . . . . . . . . . . 13 (((𝐴 ∈ ω ∧ 𝑥 ∈ ω) ∧ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦)) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦))
4342ex 114 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦)))
4443rexlimdva 2526 . . . . . . . . . . 11 (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦)))
45 eleq2 2181 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (∅ ∈ 𝑧 ↔ ∅ ∈ 𝑥))
46 oveq2 5750 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝐴 +o 𝑧) = (𝐴 +o 𝑥))
4746eqeq1d 2126 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ((𝐴 +o 𝑧) = suc 𝑦 ↔ (𝐴 +o 𝑥) = suc 𝑦))
4845, 47anbi12d 464 . . . . . . . . . . . 12 (𝑧 = 𝑥 → ((∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
4948cbvrexv 2632 . . . . . . . . . . 11 (∃𝑧 ∈ ω (∅ ∈ 𝑧 ∧ (𝐴 +o 𝑧) = suc 𝑦) ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
5044, 49syl6ib 160 . . . . . . . . . 10 (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
5150ad2antlr 480 . . . . . . . . 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 6305 . . . . . . . . . . . 12 ∅ ∈ 1o
5453a1i 9 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∅ ∈ 1o)
55 nnon 4493 . . . . . . . . . . . . 13 (𝐴 ∈ ω → 𝐴 ∈ On)
56 oa1suc 6331 . . . . . . . . . . . . 13 (𝐴 ∈ On → (𝐴 +o 1o) = suc 𝐴)
5755, 56syl 14 . . . . . . . . . . . 12 (𝐴 ∈ ω → (𝐴 +o 1o) = suc 𝐴)
58 suceq 4294 . . . . . . . . . . . 12 (𝐴 = 𝑦 → suc 𝐴 = suc 𝑦)
5957, 58sylan9eq 2170 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → (𝐴 +o 1o) = suc 𝑦)
60 1onn 6384 . . . . . . . . . . . 12 1o ∈ ω
61 eleq2 2181 . . . . . . . . . . . . . 14 (𝑥 = 1o → (∅ ∈ 𝑥 ↔ ∅ ∈ 1o))
62 oveq2 5750 . . . . . . . . . . . . . . 15 (𝑥 = 1o → (𝐴 +o 𝑥) = (𝐴 +o 1o))
6362eqeq1d 2126 . . . . . . . . . . . . . 14 (𝑥 = 1o → ((𝐴 +o 𝑥) = suc 𝑦 ↔ (𝐴 +o 1o) = suc 𝑦))
6461, 63anbi12d 464 . . . . . . . . . . . . 13 (𝑥 = 1o → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦) ↔ (∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦)))
6564rspcev 2763 . . . . . . . . . . . 12 ((1o ∈ ω ∧ (∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦)) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
6660, 65mpan 420 . . . . . . . . . . 11 ((∅ ∈ 1o ∧ (𝐴 +o 1o) = suc 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
6754, 59, 66syl2anc 408 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝐴 = 𝑦) → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦))
6867ex 114 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 = 𝑦 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = suc 𝑦)))
6968ad2antlr 480 . . . . . . . 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 691 . . . . . . 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 361 . . . . 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 4485 . . . 4 (𝑏 ∈ ω → (𝐴 ∈ ω → (𝐴𝑏 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝑏))))
746, 73vtoclga 2726 . . 3 (𝐵 ∈ ω → (𝐴 ∈ ω → (𝐴𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵))))
7574impcom 124 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 → ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
76 peano1 4478 . . . . . . . . 9 ∅ ∈ ω
77 nnaord 6373 . . . . . . . . 9 ((∅ ∈ ω ∧ 𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
7876, 77mp3an1 1287 . . . . . . . 8 ((𝑥 ∈ ω ∧ 𝐴 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
7978ancoms 266 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥 ↔ (𝐴 +o ∅) ∈ (𝐴 +o 𝑥)))
80 nna0 6338 . . . . . . . . 9 (𝐴 ∈ ω → (𝐴 +o ∅) = 𝐴)
8180adantr 274 . . . . . . . 8 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o ∅) = 𝐴)
8281eleq1d 2186 . . . . . . 7 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((𝐴 +o ∅) ∈ (𝐴 +o 𝑥) ↔ 𝐴 ∈ (𝐴 +o 𝑥)))
8379, 82bitrd 187 . . . . . 6 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (∅ ∈ 𝑥𝐴 ∈ (𝐴 +o 𝑥)))
8483anbi1d 460 . . . . 5 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) ↔ (𝐴 ∈ (𝐴 +o 𝑥) ∧ (𝐴 +o 𝑥) = 𝐵)))
85 eleq2 2181 . . . . . 6 ((𝐴 +o 𝑥) = 𝐵 → (𝐴 ∈ (𝐴 +o 𝑥) ↔ 𝐴𝐵))
8685biimpac 296 . . . . 5 ((𝐴 ∈ (𝐴 +o 𝑥) ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴𝐵)
8784, 86syl6bi 162 . . . 4 ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → ((∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴𝐵))
8887rexlimdva 2526 . . 3 (𝐴 ∈ ω → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴𝐵))
8988adantr 274 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵) → 𝐴𝐵))
9075, 89impbid 128 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴𝐵 ↔ ∃𝑥 ∈ ω (∅ ∈ 𝑥 ∧ (𝐴 +o 𝑥) = 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 682   = wceq 1316  wcel 1465  wrex 2394  c0 3333  Oncon0 4255  suc csuc 4257  ωcom 4474  (class class class)co 5742  1oc1o 6274   +o coa 6278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285
This theorem is referenced by:  nnawordex  6392  ltexpi  7113
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