r1ord3 - Metamath Proof Explorer

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Theorem r1ord3 9471

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 9456	. . . 4 ⊢ 𝑅₁ Fn On
2	1	fndmi 6521	. . 3 ⊢ dom 𝑅₁ = On
3	2	eleq2i 2830	. 2 ⊢ (𝐴 ∈ dom 𝑅₁ ↔ 𝐴 ∈ On)
4	2	eleq2i 2830	. 2 ⊢ (𝐵 ∈ dom 𝑅₁ ↔ 𝐵 ∈ On)
5		r1ord3g 9468	. 2 ⊢ ((𝐴 ∈ dom 𝑅₁ ∧ 𝐵 ∈ dom 𝑅₁) → (𝐴 ⊆ 𝐵 → (𝑅₁‘𝐴) ⊆ (𝑅₁‘𝐵)))
6	3, 4, 5	syl2anbr 598	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝑅₁‘𝐴) ⊆ (𝑅₁‘𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ⊆ wss 3883 dom cdm 5580 Oncon0 6251 ‘cfv 6418 𝑅₁cr1 9451

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-r1 9453

This theorem is referenced by: bndrank 9530 rankval4 9556 aomclem2 40796

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