r1ord3 - Metamath Proof Explorer

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Theorem r1ord3 9695

Description: Ordering relation for the cumulative hierarchy of sets. Part of Theorem 3.3(i) of [BellMachover] p. 478. (Contributed by NM, 22-Sep-2003.)

**Assertion**
Ref	Expression
r1ord3	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝑅₁‘𝐴) ⊆ (𝑅₁‘𝐵)))

**Proof of Theorem r1ord3**
Step	Hyp	Ref	Expression
1		r1fnon 9680	. . . 4 ⊢ 𝑅₁ Fn On
2	1	fndmi 6594	. . 3 ⊢ dom 𝑅₁ = On
3	2	eleq2i 2829	. 2 ⊢ (𝐴 ∈ dom 𝑅₁ ↔ 𝐴 ∈ On)
4	2	eleq2i 2829	. 2 ⊢ (𝐵 ∈ dom 𝑅₁ ↔ 𝐵 ∈ On)
5		r1ord3g 9692	. 2 ⊢ ((𝐴 ∈ dom 𝑅₁ ∧ 𝐵 ∈ dom 𝑅₁) → (𝐴 ⊆ 𝐵 → (𝑅₁‘𝐴) ⊆ (𝑅₁‘𝐵)))
6	3, 4, 5	syl2anbr 600	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝑅₁‘𝐴) ⊆ (𝑅₁‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2114 ⊆ wss 3890 dom cdm 5622 Oncon0 6315 ‘cfv 6490 𝑅₁cr1 9675

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-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680

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-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7361 df-om 7809 df-2nd 7934 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-r1 9677

This theorem is referenced by: bndrank 9754 rankval4 9780 rankval4b 35249 aomclem2 43486

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