leweon - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > leweon		Structured version Visualization version GIF version

Theorem leweon 9933

Description: Lexicographical order is a well-ordering of On × On. Proposition 7.56(1) of [TakeutiZaring] p. 54. Note that unlike r0weon 9934, this order is not set-like, as the preimage of ⟨1_o, ∅⟩ is the proper class ({∅} × On). (Contributed by Mario Carneiro, 9-Mar-2013.)

**Hypothesis**
Ref	Expression
leweon.1	⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}

**Assertion**
Ref	Expression
leweon	⊢ 𝐿 We (On × On)

Distinct variable group: 𝑥,𝑦 Allowed substitution hints: 𝐿(𝑥,𝑦)

**Proof of Theorem leweon**
Step	Hyp	Ref	Expression
1		epweon 7730	. 2 ⊢ E We On
2		leweon.1	. . . 4 ⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}
3		fvex 6855	. . . . . . . 8 ⊢ (1^st ‘𝑦) ∈ V
4	3	epeli 5534	. . . . . . 7 ⊢ ((1^st ‘𝑥) E (1^st ‘𝑦) ↔ (1^st ‘𝑥) ∈ (1^st ‘𝑦))
5		fvex 6855	. . . . . . . . 9 ⊢ (2^nd ‘𝑦) ∈ V
6	5	epeli 5534	. . . . . . . 8 ⊢ ((2^nd ‘𝑥) E (2^nd ‘𝑦) ↔ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))
7	6	anbi2i 624	. . . . . . 7 ⊢ (((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) E (2^nd ‘𝑦)) ↔ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦)))
8	4, 7	orbi12i 915	. . . . . 6 ⊢ (((1^st ‘𝑥) E (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) E (2^nd ‘𝑦))) ↔ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))
9	8	anbi2i 624	. . . . 5 ⊢ (((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) E (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) E (2^nd ‘𝑦)))) ↔ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦)))))
10	9	opabbii 5167	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) E (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) E (2^nd ‘𝑦))))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) ∈ (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) ∈ (2^nd ‘𝑦))))}
11	2, 10	eqtr4i 2763	. . 3 ⊢ 𝐿 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (On × On) ∧ 𝑦 ∈ (On × On)) ∧ ((1^st ‘𝑥) E (1^st ‘𝑦) ∨ ((1^st ‘𝑥) = (1^st ‘𝑦) ∧ (2^nd ‘𝑥) E (2^nd ‘𝑦))))}
12	11	wexp 8082	. 2 ⊢ (( E We On ∧ E We On) → 𝐿 We (On × On))
13	1, 1, 12	mp2an 693	1 ⊢ 𝐿 We (On × On)

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 ∨ wo 848 = wceq 1542 ∈ wcel 2114 class class class wbr 5100 {copab 5162 E cep 5531 We wwe 5584 × cxp 5630 Oncon0 6325 ‘cfv 6500 1^st c1st 7941 2^nd c2nd 7942

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-int 4905 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-ord 6328 df-on 6329 df-iota 6456 df-fun 6502 df-fv 6508 df-1st 7943 df-2nd 7944

This theorem is referenced by: r0weon 9934

W3C validator