r1ord3 - Metamath Proof Explorer

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Theorem r1ord3 9765

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 9750	. . . 4 ⊢ 𝑅₁ Fn On
2	1	fndmi 6637	. . 3 ⊢ dom 𝑅₁ = On
3	2	eleq2i 2852	. 2 ⊢ (𝐴 ∈ dom 𝑅₁ ↔ 𝐴 ∈ On)
4	2	eleq2i 2852	. 2 ⊢ (𝐵 ∈ dom 𝑅₁ ↔ 𝐵 ∈ On)
5		r1ord3g 9762	. 2 ⊢ ((𝐴 ∈ dom 𝑅₁ ∧ 𝐵 ∈ dom 𝑅₁) → (𝐴 ⊆ 𝐵 → (𝑅₁‘𝐴) ⊆ (𝑅₁‘𝐵)))
6	3, 4, 5	syl2anbr 611	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝑅₁‘𝐴) ⊆ (𝑅₁‘𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∈ wcel 2145 ⊆ wss 3899 dom cdm 5655 Oncon0 6357 ‘cfv 6533 𝑅₁cr1 9745

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7417 df-om 7864 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-r1 9747

This theorem is used by: bndrank 9824 rankval4 9850 rankval4b 35608 aomclem2 43897

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